Advanced Engineering Mathematics Solutions Manual

'1.

= == =

H H H

Section 3.9

(b)

31

~~~~~~~dt., ~/V'A.~· X,(o)= I= G~
~(o)::O= G~cp-H~"I' x:toi :: o ::: G<-04' + 43 Hee."'

G~4>= 1/2 a.o. ...i-. ~--+ H~"V:: 112. X~(O): 0 = Geo:>cp- ,(3 Hec>'I' c; Cc>4>:: 0 4'3 HC(.) 'l' =o Ao x,Ct) f ~l.t+ Tif2.) +f ~(,[3 .:t +1T/2) ~ <:.r.:>-t + tCf,:),[3.t

=

X 2 lt):

t

~(:t+TT/2.)-

m~~

~

c:P= "o/= 1f/2.) G=H= 1/2.

=

i ~ {"3:t+1f/2.): tC/:):t-t Cr.>"3:t

P.eot~~~tk

~~~

~(~):

~({'X,:0.5~ Ctr.lC:l:) + o. 5 C\1J(~Jtt(3)~ t.)) x. =

*

-0.5

O.Sf Ct:>lt)-o.st:~(Tt(3)~t)} t= 0 .. 20, x, = -2 .. 2 ,

~=,CXXJ)j

Section 4.2 CHAPTER4

Section 4.2

5"

-7,

(f)

\

x,'2:.3x,+ I ('X.+2.)2.. - -

~

/

R=S"

-,'\

\ --iK---+---+--.-. I

3 \

' '-

~/

/

'

....._.

___

/ .,,,.

/ /

x

---- R=2

34

Section 4.2

35

Section 4.2

36

Section 4.2

A:

37

= Bex- xz.+ _ix,3 -tx." + ,~ x 5 ) + Ocx.~)1 ~ ~~ OWl. ..n.-.J.:t,.;,., E)t~ '7Cb) (~ A..o ao 11-d B..:a o..,12.). ( :f) Fcfl-: 'X!'~"- _"j =O) d1.t. ~~&.. ~ ( 'X."2.~ cl.4tf"(1('X-), 'X-,~ )-~(X) ::O, :at')(),~ =~)j .~ ~ ~ ~~ oJrnt ~=O. fa~~ ~p~ ~ .

'd(X-)

x =2, µ 0

~ -r=~-2- ~~ODE~ (~+2.)i:Ycr.)-°YC'?:)=O. N~J4

C6'W\'VW\~~

~( (~+2)"'2 -1 cLU,(Y
=o, Yee), ~~~)j

~ Yee)= Yeo)+ J){Y){o)r +~"!to)r.2-+ (-fcr. Yto)+ g !XY)(o)) "i: + ( i1sq. Yto) - ~ J:XY)(o)) %: ,,_ -t Yeo)+ 1 ~20 DO')(o~ cs+()(~') 3

8

O'l.J

(-clo

i,...W~; ~C-x.)= C, [1+ ~('X.-tf-iq. (x-tf + ~ ('X.-1.)4 -fiz, c-x.-1f] + C2.[c~1)+ -f4Cx.-t.)~-fs
13.

"a"+ 'J =0. d ::< l\o + 41 'X. + /J.2. 'X..,_+ £\3 x_3 + a.
C,£l3X + l2.Ct4 'X-2..+ 20 Q5X.3 +···)

~o; 2~2. + Q. 0 =0 ~I: '43 + t\ 1 :: 0

__,,

a.i == - a.o/2.

r

Section 4.3

38

+ (q 0 +q, 'X. + 42. 'X.-z..+ 4s'X.3 + 4q. %'f + Q5'X.,S"+···) =0

-» Q3: - ll.,/G, X : 12Clq. + Q2 =0 ~ Aq. -£t2./1'2. O..o/2.4 3 'X., : 2oa 5 + 4 3 =o 4 a. 5 = 3 /20 a. 1/120; 4 ~. -rk lfA>l) _r__ N\fx) =Ct 0 +lt I x - Ai>x-z.. - T4 • x.3 + Aax. -tAL 'X,c; - .•. () 2. 2't 12.0 4 =llo (1-J.. 'X,,,_+ $X. -···) + ~. (X.-t; ~ 3 + x_'ii_ ... )

=

=

2

=

-a.

IS.

r~ E~~s,

_LI x -

=l+~+~'Z.+···+~n-1+ .:£::...,~Sn= l-'X. Sn

Sa+···+ SN _ (l-X-)+ (l-'X.'2.) + ... -+ Cl-~ 911 ) _ N N(l-~) -

:-1- - 1 l-~N 1-X.

N (I- x,)i.

_c.n -i'"''"

-z.

N-(~+'X.: ,_ ···-+ 'X. N(l-'X-)

N

clo

·-~n. ~} J~

) : _L _ ~ \-+ 'X+· .. + X\.J\-% N I-~

jQ X.:Fl. ~- l'Xl
J

1-x

1-X

l

=J_.

1-'X

cap-1~1>.1 ~ik ~N~oO a..o ~~ oO D-Mtl.~ ~ ~ ~ ~ ~ ~·

Section 4.3

Section 4.3

39

*

NOTE CAREFU~L)': ~ ~ ~ ~ ~.~~ 1-0 Ex~t.~2./7f, fen . .t?.~. HAl\(W\' , ~ ~ ~ ~~ )M~ ~ ~ 2.1\tl ~A 3.J\
~. ~'14 ~~fa~ (l.e., Z:n) kvv~ ~J wt~~~ ~M.~ ~ to .k ~ r.-.-t, ~ ~ -tk -1 a.-tA -2 ~ ~ t; o. W.e. ~ ti<>~ ~ ~ >1=-1 ~ ~ Z_ 1 tl{Vl+l)Yl~~+l'Z.~ ~~~ ~ {Jk V\+1-fo..c1dt;

~~,ik n=-2~ ..-.=-• ~ ~ r::_4(t\+2.)(r'l+l)Q.n+2.'l-~ Mt. ?fM ~ ~ j tk n+2 o.,..d. n+ 1 ~. .0... +'-t ~ ~, a..4.tl.~ .w< cam ~~ ~ 3fo ooaAM. L3 (~-? ~ ti._ 3 ~z =o._1 =o), kt~~ thL -21<, o ~ ~~~AN\~~ Mt~ ~e.NJ~ n=-2AMtt t\=-1. ~,Jo~~-~~~ ~t-, ~~ '2_~ "-
hi<.4+, =

z:_

L.;' ~

r.: ~ ~

~=t!.,,_=q3=~2=~.=o. tk, ~ ~

~

n+~

L_2 [
/.l-0

n : -2 :

Jt. (51-2.) ll

=0

~

'f1~'f.3.I ~~ ~ n...,t

JZ..(Jl-2.)

(Yl.=O)

=O)

=0 ,

-~ n =-2,-1, o, I, ... Jl =D, 2.

.w&.l ~ ~~

dl

~'~

dk~ (12=2.) ~ ~ ""- ~-,J4 ~ ~ ~ "';J#•c.).J;t~-tt-..1 '1.=0 ~ i'< ~ A.t..~4.;) .e.:t M.o. ~ ..:t ~ ~ ~ A
w

~~ ~ (n~2.)na.n+z. +4t.ln_3 n= -2.: ~ ao= o.n.er.

14

-a, :::o ~ a1 =o n-= o: o=o M> ei,=aner. n= 1: 3a. 3 =o ~ ~3 =o

n= -1:

Btt 4 =o J>-O llq. =o n 3: 1sa.5 +440 o 40 l
n= 2..:

=

=

'ttt)

=o

r-WJt'Nt ~ ~ n=-2..

{n=-t>o, 1,2 , ... )

n=s:

35(.{.,

+ 4-l(,a, =o Ao

n=b: 48a.9 +4a.3 =o

AO

n=7: b3d,+4£4=0

M)

Q7

= -fs-a.2.

a.8 =o

q,=-tsa.q.=O

n=s: goa. 10 +4a 5 =o ~ a 10 =-1z,t:
=Q + Q,:tz.-ia ts--fs: £lz.t7 +;/s~t 10 +; 5 Q.2.t + ·· · 5 7 = ~o (1- ~t + 7'5 t' + a.2.(ta-~f +1:.. 5 35 gs t'2.- ... ) 0

0

0

- ···)

1 2.

.$

Section 4.3

n=o:

a1 =o

5z. Cl..3 --o

n -1L• - ..,... •

'-'5 -

n -5· - .

n~-..L"'

/1

~

-

_l

- "%. 144 it:,

-

I 2~'Z."'Z.

a_

0

40

Section 4.3 ··'' .~o

[~~ 1 -1\1"(1 1 ]k 1.t

rfi,

+Lio0 n-z.Cn~n-1 -2: o01 Cn?ln+l =-2~;


41

=-2. ( t~ + -k-X. + · ··) 3

x0 :

C1 =O L#Cz. =-I AO C2. =-1/4 'X,i.: ~c3 - C 1 =O MJ C3:0

X:

X-3 : lbCq. - C,_

'l'~}

N~TE:

=-1/S

AO Cq.: - 3/L2'i1

>

AN-.d

1\12.lx):: M l'X)~~ -.Lx,2. - ~ ~4- .... ('Ll.lu ~1 . A-- -;- nJ, di 'X.~ '+1·f- 4%d ::o12.8 AO -,~ ...i.J T"t. ....

1

A
(')

A

:t .

·~ ~·"~c

i

~ 0)

~ ~ ~ 'd(X) = C,IOCX.);. c2. Koc~)) ~ Io, Ko a.n.t. ~~ ~ ~I;. o'lAut 0. 4...J.. t4 ~ ~ ~ ~> ~J 44' ~ k_ . ~ ~ s~ 4.~. C7.wi 'd•l") Ao IolX) ~ OW\ 'JlX> .AO a__~~~

i

~ I

0

~ K0 •

NOTE: Before continuing, note that the different cases defined in Theorem 4.3.1 depend on whether the roots of the indicial equation are equal, differ by a number not an integer, or differ by an integer. Since we will not be solving all parts of this lengthy problem, let us at least give the indicial roots for all parts of the problem, to assist you in choosing which parts to assign. (!) r = 0,1/2 (Q) r = 0,0 (c) r = 0,0 (d) r = 0,0 (S) r = -1,1 (f) r = -1,2 (g) r = -1,l (h) r = -1,1 (i) r = 0,1 G) r = 0,2/3 (k) r = 0,1 (1) r = (1±-{3)/2 (l!l) r = -1,2 (n) r = 0,4/5 (o) r = 0,1 {P) r = 0,1/2 (q) r = -1/4,3/4 (r) r = -1/4,3/4 (s) r = 0,0

Section 4.3 (YL'2.-3n.. )t{>\

=

(V\::t,2., •.. )

(t\-2)49'-l..

n= I! -3Q. 1 =-4.0

=0

~

't 1 : 4o/3

n= 2.:

-2Q.i.

n:3:

oa3= a.,::. 4.o/3 ~ 4o=~ ~,Ao. A.~~-~J

AO l.l2.: 0

_

.

.

Jl.=-1

W:t:>~~--

S.t:t

(~ ~ ~

st=2.

(n.-z.+3n)~n

n=l:

=(n+l)«n-1 .

4a, = 240

M:1

42

~~

*' ~ ~, ~ 1o'P~4.'3.I).~*~ (Y\:(,z, ... )

a,= a..o/2.. a.2.= 3Q.0 /2.0 a3 = a0 / 30

n=Ll-: ~go..4 = 5"£i3 Ad ~4

= l{o/l"s>

n=S": 40as: c,a.4 ~a;-:: a0 /U2.0 n=3: I ga.3 =4tl2. M it:,· m1 _ . '..L~ ~ o0 t\+2. t.. J.. 3 3 4 .L ~5 ...L t;. , .., "f\.WJ. '~ ao=I ~ J (X.)= ~o '4'nX. X -1- 2. ~ +20 X. + 30 +T,8 ~+ii'i:Ox_+ ... To~~~~ "J2.(~) MAt (41c): ~ 2 (x)=: K~ 1 lX)~X ~ -x,-• dn~". n=2: 10~=

34. 1

00

I

=

fd1

~~ ~

.v:.:to-f4

L.a;

ODE ~

x. [~-t 2K~:1x -K"c)1/X't.+ L~(n-l)(n-2.)GtnX.n- ] rH 1 -x.~[~+ Kfil'x.+ :L~fl\-l)d."~n- ]-2.[~x+L:tl,,x. ] =o 2

dt, efl.)

3

= =

1

L;'Co-1){t'l-2.)~.,,xn- -L:(n-1)cAnx,n-2.L:d.n'X.n-• K('d 1 +~~.-~x~r) 1 3 3 (~i +2.cl3 Xt.+ 'd.q. X + l2.d.5X. 4 +··-) !K( 'X.t.~ f~ +:0- ~'f +··· -( -d.o + c:l2.'X.'Z.+ 2.d.3 x 3 + 3 d.4 /(.q. + . . . ) + x3 + -£ x.4 + ... 3 3 -2. ( ~-· + ll, + tA.2.~+ d.3X,z.+ tlq. x +dsx.c.f.+···) -4-x.'2.-3X. -t~lfX-0 :
... )

=

X1: -2tl, 0 M) ~2. =0 X.-z.: ~-d.2.-~3= K(t-4)J ~ dz.=O, AO .wt.~ K=Oj ol3 =anlr. x.a: t;,t1 4 -2d..3 -2.&.q. O A-O olq = J..3/2~t: 12tt 5 -3J 4 -2~ 5 o AO ds =Sclq./10 3d..3 /20

= =

=

.t.k

~

dzcx.) =o ·:V'X'.l t.-'lC + '<'.-1( Ao+~~ +ox.z.+d.3~3 + 4:1"'4+3,tJ x.s+ ... ) = d.o (~ +t)

i,x."' +· .. )

+ tf3 ( x_2.+tX.3 +

2.0

H~~ ~ ~ 'dlX)} ~ ~ ~t ~

~~ ~

'jz.00 ;le,

LI~. ~'it ~t ~ it.e c.t3(X~+fx +.1..'X..'f-f..···) ~ -Arnk · 20 ~ hfu.ro, ~·· I , ~ d 0 =1, ~' 'd2fX) ~ + t . o0 Y\T. Cj.) 3X."j" +:'j'+ "c1 =a. '.()
=

~ dt,

/:l-0

n= o:

+

11

~ oO "'- 0

[

2.:

-+

" ]

n+n.-1

3{n+n.)(f\+'1.-l) a..n. + (f\+Jt)lln + ltn-• ~

2

=0

L: 00

a

a

( £L1:: 0)

[ 3(n+.n.)"2..- 2.(n+Jt.)] lln. + 4rt-l == O -fO"l n = 0,1,2> ... (3Jl..-z.-2J2..)ltc, 0 ~ .Jl..= ~' 2./3 /M\c! qo = an..fr. E~ .n ~ ~ ~ ('t\.t.. ~.

=

n-1

xY\+st-1 -0 -

Section 4.3 F~ .ti:' .Yl=O: ~

a1 :

Yl=I:.

*~

>1=3: 2.ll.l3

n:: 4:

= -a.rH fo't n= 1,2.> ...

-Q 0

aa.2. = -a,:: ao

n=2.:

(3fl-z.-2.n)a.n

43

=

a.2. = ao/8

Ao

lt.3= -~o/1'-8 a.o I I" 8 M) Q4 = a.o ; , '72. 0

-Q2. =-a.o/8 A.a

40a.4

=

= - a.. 3

~

Ac,~a 0=1~, 'daC~) = I-~+ ~~i. -i!a x,3 + '~2.0 ~4-- ••• NN;lr, .L.:t st= 2./3: ~ ~ ~ (3n.-z.+2n)an =-a..n-& -ftn. n= 1,2>···

sa. = -a.o

n= t:

M:J

a,= -0..ol~ =a.a /so

n =2.: 1'a.i. =-a 1 M1 'l2 f\ = 3 = 33 a3 =- a.2. ~ a.3 = - a.o/2.(,40 n=4: S,4_4:: -q 3 A<1 ~4 = a 14'7~40 0 /

.J.c,

M)~ao=I~ ~2.(X,)

(tn.)

=

n +2J3 1A.n'X 0 o0

=

2/3(

X.

J_

_L

i..

I

3

I- 5" + 80 X.. - a.e.,40 X:.

+

1

4

141g40 ~ -···

)

X.~"j"-(2.+3X.)~=0.

"J(X.):: 2:~t:tn~Yl+Jl.r oO n+Jt+I L~ (n+n.)(n+Jt-1)4'n.~n+n _ L~ 2.Q.n.x,n+ -3L 0 £:{nX. =O

en ~

r: cn+nxn+.n-1 )t.tn ~

Y\+n - L.~ 24n x'f\+11.

L.~ 3 a.n_, ~ n+n. = o (1'-tS7.)(n+st-l)-2Ja.n - '3a.n-l} x,n+:Jt = 0 (a_,= 0)

r: {[

[Cn+Jl.Xt'\+st-l)-2Ja.n-3a.n-i

Ao

2

n~o: (Jt -~·z)a 0 =0 ~

Fu~~t}U n=1: n=2.: n=3:

-

= O fo't· n=o,1,2,... ~tA ~o=anlr..

.n.=-1,2.

*

{~;;;J-{ ~m~ dt~).~°*~ cnz.-3n)an = 3lln-1 fdt n =1,2, ... -2a.. = 3llo M) a,= - 3do'2-2ct2. = 3 Q, P-0 tl2= -S0.., /2'tto/4 . . Jl=-1

oa3=3a2

=274o/4

=

~ t:{f)=o ~~a..~.~) n=-1

~ ~ ~- 1\JJ. ~ ~ ~ ~ ~ n= 2 ~ i4 otlut ~(I.fie).

n=2: tk*~ (n'2.+3n)a.n:: 3lln-i n.=1: 4a. 1 3a. 0 A
=

·

-/di n=1,2, ...

=

=3ll1 /to = "J0.. }40 n= 3: isq3 = 3ll, MY a3 =a2.I" :: 34 /ao n =4: 2.8 ({4 =3t<3 '°° a.q. =3Q. 9 = ~a.al 2.240

n=2.: lOQ.2.: 3Q 1 A
0

0

3 /2

it:,.

.

~ > ~ ~o= I,~' .

'fo ~ fk

.._,o0

n+2.

"ja(X.): Lo~nX.

'2. 3 3 ~ ~ 3 5 ~ ' : 'X, -t- 4 X +40 'X. + BQ?C +fiiro" +···

~d ~ ~'2.C~) ~ {41c..):

'diX) = K "J/?l) ~'X. +

r:

d.n'C,n-a.

P~~~~ ~tkOj)E~

Section 4.3

44

Section 4. 3

~ Jl~> = x.:~·-+i.f. 2:: o..n.-x," ~

10. ca.>

'O''+r~+t:J

=f

o..

45

:t. ()"+ti:i'+ i-a=o, ~ o =

~

+r1·+~i ~~ ;-10o. ~ M c...tl ::Cl."~z:a.,,x.."

a_n'X.'t\. :: X. ~Lo; ~x,""J ~ AN-t. Ac.t. ~(!-'fl== 'ln. iin.::~.,. = c.,.-AJ... , a.....J. lt~.'2.) ~ (IO.I), J.:t......,: 'd('X.).= A'X-"<. [ ct::>C~k~) +A~ (@~'X.) 1L ( Cl'\-tA.dn )~~ 11 +Bx. cc [ " ] L. (cn-icll\) ~n

= x,ct-.c-'f L:o Cb) p~

= x,« [ A~o !: CnXl'\. ~ A.t ~) !: cl.n~Y\ -A~c >L ol.r\-x. +A~~(

)L

n

Cn'X.

+Sepe) !.cn'X,n - B.A,cp() I:clnx,-n. -B~c) L. d.nx,n -B.t~o !: c.Y\'X.n}

= 'X,clf (A+S) [ er.>c? ~Cn.X-n ~

(c) X}·Nj.~-+ X ( \-r't,)rf + ~

.

rr~ .vdt

~c) !: dn'X."'}

+ ~ll\-6) [Col l 1.J.n?C." + A
= 0. Jt~-.n.+.n + t = 0 ~

.n= :ti.

AO

1-i'k

ol=O AN\4'

~ =~

~,~Lt~~~ C~[ 1 ~ ~ f~; fdt ~' ~ AAnlQ. ~ ~c~~):c. ~'1 ~(~~K.,):s. S.tt..k ~(X) = LooO ( C.Cn - scln) 'X.,'I\. • ~ 1f'X.) =L (-sct\-c~~~V\C.Cn- Y\Sdn )x."'-l "ti"~) = L [-c.c..i.+ Sdn-VISC.... -ncd., +(tl-1)(-sc.... -cdn +ncCn-n.sd.,)] x."'-2.

..k

?~~~-rkODE~

'X~,,~ +~'~ +N\~

L~ { C [Cn-z-~-j')cn -(2n-\)d.nJ + S [-C~n-~cn-(1'i.-/1')cin] .+ C..(ryin-~n)-S~+~) + c(d~)-S~)} ~n

0

dt

'f"

oO

n+I

+ 2'0 [c(ncn-&.n)-S(ruln+Cn)]~

+x~'~

=O

) rooOf c.(nt.cn-2.nGl.n,)+ s (-2.nCn,-n'Z.dn)JJ?tY\ + Lo[c(nc~cln)-S(nd.n+Cn )]X.Y\otl = 0 0()

~,

cit,

~

L 1t

~

1X.n-+L1 {c[{tH)Cn-ici"_11- S fJn-l ~n-a"'° ct\_, J}xn =O

''

.0

I. fc[nt.cY\-2.n.d~ +(1'-l)Cn-tdn-11 + s [-2.nC~-n?.ci~ -(n-l)d.n_,-cr\-1]} x..Y\ = 0 ~

T..lu. ~ ~~ AO~ ~At:I: •=O ~tlr=O:

n?..Cn-21\d.n + (1\-l)Cn_,-d.Y\-l

AJtt.

*

~ ep(.D..,x) 't-n a-.!. Ai,,.(J.,~) -x." ~;

=0

-2nc"-n-z..oln- (n-l)~n-1Cn_ 1 =0

.

,fort ":=1,2., ..• , ~ Co~ c\ 0 ~ ~ • n=l ~

c,= f-2.C

+a0 )/S

=

~a d. 1 -(co+2.d.c,)/.S'" n=2.~ Cz.= (2.C.0 -c!o)/2.0 '1,.nd d.z.= (Co+2.tl.0 )/20

n:3 ~ ~c:l.

,Q..() 6V\. •

C3

0

=-(\'1C 0 -,d.o )/780 ~d.3 = -

('Co+ l1d.o)/780

Section 4.4

46

~~~cnj"~~~ ~~(10.3)~~~1-4~~ ~ C,D,c;,,cl0 • Lit~A.tt ... W..t.~ ~fk Cvi''°' ~ aln'AJAOJ ~ C=1 ~ .:D=o ~, cto.3) ~ "a(X.) = Ct=>(-k 'X.) tc0 + ~(-2.C 0 +c:lo)~ + k,C2.C 0 -clo)~i.- 7180 (17C 0 -'9d. 0 )x 3+ ·· ·] -~(~~)[ cl0 - ~(C0 + 2.d0 )"-+ia (C 0 + 2.d.0 )X.?..-ko ('C0 +l7iAo )'X.3 +···] J

=Co { ~(~'t) [ l - ~"' +~~2.-J:z.. X-3 + ... ] s 20 '780 } -~ckx)[-.L x.+-1-'X,i.--'-- 'X.1 +···] 5

20

7SO

+<4 { C4:l(Jn~)[ ~X.-:$ x,i.+ ~ "3_ ... J

-~ckx) [ 1- ~"-+Jo x..,.- ~~ x.3 +··· J}

NOW"" M CAM. ~ -tf-.4' ik ~ 40,.tM ~' ~1 C-.:t D.

Section 4.4

1t'-<. ~

(10.3),

~ c.,Jo

h

~

Section 4.4 (n+i)~HC~) = £2n-r1)~fn.<~)-nfn_ 1 c~)

dt.

at '7. (O.,)

47

Cn=r,2, ... )

nPn.c!(..) :: C2n-1)~fn_1 c~)-Cn-l)fn-2.Cx.) cn=2,3, ... ) a/ax ~ -.L -2rz. = "'£ o0 f. '('6) SLn 0 -

2.

.

(l-2.XJ?. ... Sl'2.)3k. n

oo

(l-2.X.lt+ Jt'2..) 312. n+l oo

0

=

2:

n.

o0 (l-2.i(.Y(.

+ Sl..'l.)t>~ ('X.).n.n

0

=L P~CX)lln - 2~ L.oo P,:cx.)Sln+I + L eo Fn'<~)Jtn+2. L7 Pn-1("-) Sln =L~ Pn,'cix.)nn- 2~ !,o0 P~_ 1 l"-)J'l..n + L.'; P~-a. -fori n =2)3, ... 'Pn-1 l~) =p~ ('X. )- 2. x, p;_, ('X.) + p~-2. <"-). -R:

L0

~(~)Yi.-¥\;-

0

0

0

('X,)stn

.AQ)

8. (a.) ~ ~

4

,,..0.

1o

"1-p~

-t~(~:~) =-k 2.(n-+!-3+~+ ~'1+··) "'oo ""2n

=2. L I

~

O ~n+I

2.

> AO ----

2.0+1

=J

l -I

fort \n.\< ~ f Pnlix,)l1t.. tJJ'i

,,_f:O

•

1-

10. (Cl.) Cl-'X.2.)d-''-2x'd + ?"()=o. To kt~~ ~=• ~;,a,~~~ 1l-1:t. r~ ~ ODE ~ tC2.+t)Y''+!i.O+tff '= o -':J~ Yrt). rp"'- ~ ~ ~ Jl~=o Ad .n=o,o. 1JJ!... n=o, ~ 'frt) ~a.ntn ~cl ~ (2.t+t2. )(2.42.+'43t + ... ) + (2.-+ 2.t )(°' l + 2~z.t + 3ll3 tt. + ... ) 0

=

=

/.)() 'la :o

;t.O: .2.tt I : 0

= =

iA 2:+ 20., 0 ~ a.z. =O :tz.: I 8tt3 +'tlz o tK> a3 =o ~ M) li'f\· ~) Yrt): t.l 0 +0+04-···=a.0 MJ ~ ~ fort). ~d :IO (41b) .W..

t' :

~

Ito-

Y.
Y2.(t) = Cl).0..J: + ~7 Cntn • P..d::t..j ~ ~t4 ODE ~

(2.t+;tz.)(-iz_ + 2C2.+,c3 t +···) + (2.+2tX f + c1+ .2Ci.t +3c3t'Z.+ ···)

£ 1: t0:

-2-4-2. =O

-I +2.C 1+ 2: 0 AO

c, =. -1/2

:t: 4C2. +4c2. + 2.C 1 =0

Ao Cz. = l/g

=0

£t().

0..-+ oO .

Section 4.5

50

Section 4.5

51

Section 4.6 60~0. 'f~, TC&o)~ 2trf17i

>~

~ ~' ~i{..i t!/til:

-e+

t~e=o.

52

o) ~ 60~0. NOTE: PJ\~d ~~ ~

1-

(1'R.1)

~

~~ ~ ODE

.

.

.

.

fdt- ~ ~ (~.e. 1 J&l<<.1.~), ~e-e ~af-4 ~ ~4

*AA.~~ (S~ss) ~~~ -e-+c~/i)e=o,.w4 ~~ Sft)=,L)~(~.t+
l,.(b)

ro't

f
Cx.2.

(aoX..:JO).

W..Jl) fC't)-4 = -q.~z.+ qx
·· · rv - 4X'Z.. ~ x_,o . ../ I' 0 c~> o..o. ~...,.o ~ ~ to- ~ i-kt Hr~> A.J CX- ~ "'~ oo. tJ.J.l,

=

cf) fort, H(x.) Hex): :ix.'3-x.-+l x,l+4

=ix ~~"'o0. v

7'X}

"J

x_"Z..

Section 4.6

+

2.. ~)~ ~ astt.~ ~
[ W3/2.) _L_

+

(2)-(10).

L.tt°MO. ~~(')ANA

{JO)

c-rf! (~)'2. a (~ )4 ] rCS"/2.) 2. + .2 l1(7/2) T + ... '2.

=~ftrn-t(tffe) ~ +2(-H~X-Hif) ~ -···)=J;( 2 -tx~+tc;?l -···) =,_fi'X rr (l-zC , +~Q.-···) 120

4

4

=~ :z... (~-z!-+-~ -···) = r:!:" MM"'. rr~

s~

3!

~~

S~~ {o'i J_Yz. C'X-). 3. "foe):)= rr [ ~
+ Os'-kz)~~ C/:)(x.-~)}

=#L~~~-~?l+Jrlr-11;&CA~-~+r +#¥2)~

(~c.r.>?G + t~?C.)}

= ~ (~~-er.>~) ::: ~:x. ~(?C.-%) > ~ Wf. 1\tt ~ e<.>tA-8)= epACllB 2

+~A~8~~fA-B)=~Acn8-~ fo't. iL. CA4t. ~ '1: ..6 J. 'T'W ..0, W<. A.ul:tii ~1".ct:cr-iA

rrx

4. (
(4.1) (XVJl,('X.))': X,VJV-1(~)

~ (') J

x, JvlX,) y

M

(-1)"-

= Lo -k! T1Cv+,k.+a) 00

11

,

00

2.(-k-tvH-n*

l(.i.h2v

2

2t+v

Ex- J/x.>] = Lo -k! f(1'+k.+1)

~2k+.t1'-• 2.2.fi+v

co

c-R.1-v)(-•r

~2.-k+2."-'

=Lo -k! rev+*.+ I) ~il+v-1 *

Section 4.6

53

Section 4.6

r_:

55 :

r.:

o'(l~ ~'.. ~.Jn.(X.)tn.+ fJ 11H(t.)tn =r::(n+l)J°n+l(X.):tn ~~~~~1~t~1:t ~ . fJn('X,) -i-fJn+2.("-) = (n+l)Jn+l(X.) cY?_)

r~, rs.3).

ctJ.,) A~.> (~2) ~ ~ ~

~ it_~~ ~ {tn n=I, -~~.t~x.) = :!_ [J0
~ n=o

~.

J.'l'X.)

J(l,'~),~)j

.

= J;,C't-)-~J.C"'),

~

(8,3)

1di n =o

.

kt it~ 4.-h -

!~ ~ ~~ ~ ~· Hl\V
=2.\CX.> - ~C"-)

J°2
Y'I\+

J;tx>)

~ ~~~ J;. IX)) ~( J,~x.) . ./

~I l?C) = HJ,, (x)- ( 2.~X l -

J :::

')(.) -

6W'\.

~~

ec>(ll'..~~)d.& = TT L ~=o ."" c-~>~ x.2.j f.n.o.;...~e-.1e (2~)! ~

....L J.11" 1T o

_i_

• r 4ff'J "· ?.~. • 2.a ed.-0

a

~= 2.J

0

= ~(~)~( =

r(¥)

2

t1

J

rct)

rca+I)

~

k-

J

(~ii..e..~L i.o,~ ~ f7=1ll2.)

1? (".1) ~ Ex~I' 1 ~'4.S 11 (14.1) ~ E~~ t41 S~4.s

°t')

=Ji!. rr~+4:)= J!f <1-f){~-!)···(:.L)~r~) =.U. c2~-1>c2~;3)···
= ~ J_. "l· 2..°I

0

d

2.

c2~-1 >C2;-2x2~-3)·· ·C:z.XJ)

= ir

c2j-I) !

~!2.~ 2~-Tc~-a)~ . M .L f.fT Ct':>(~~e) d.-0 = ..L (-&)~ ~'-j rr (2~-1) l _ t T_6? (-l)~ 'X,;2-t_·....._,;!('_____ n 0 . 1T ~:o (2~)! a~2.~2f-'(~-I)~ -'j( ~=o 2~ ~\2a2~'" 1 (~-I)! (2l-2.H2a-4) .. ·2.

.

lo0

~ )~ (X..)2.~ =L-a=o CjD2. T oO

10.

(-1

s~ ~ a"1"

(41.i)>,c)

OQo =0 ~ 4o= t.=I: (q-'3)tt 1=0 ~ q1=0

-k =o:

'k.=2.:

=

./

l~ 'lJ=3,~, ~~ ~ Jt=-v=-3.

an.er. = -ao/8 a 3 =o

(1-~)~z. + {{ 0 ==0 ~ 1{ 4

-k.=3: co-,)a. 3 +a.:o M) 'k=4: (1-~)a.q + a.2. =O A<1

*-=5:

J()lX.)

C4-~)a. 5

+ a. 3 =0

a.4 =-Qo/'4 MJ a5 =o

oa,+ {{ 4 =O ~ a,=~ ~tt ~q.~o. B~ ~=o ~ ~ii4'

i='1:

~~ 42=0~.~~o~~~~. --t=1: (1"-')a..,+ a5 o MJ a. 1 =o -ft.= 8: c2s-·n ct 8 +ct"= o MJ l( 8 =- a., It'

Q0 =o

=

~.

MJ 'O{?C)

-i.-t.Jl ,oa -k-3 -3 -z. 3 4 Q S" ='«> Lo Q~X :: Lo Qi~ = O~-t-OX,+0~+0-(+0X-+O~+a.0~+0~ --rt~+··· =a.o ('X,'3 - -k; ~s-+ ... ) -t

o

i.

Section 4.6

57

1

15". ~''+4~ =0. Q.,=0, b=4, c~o, ol=i,'V=l/2. A
=A~.~2x.+Bfff cr.>2x =

C,~2'X.+ C2.C1=>2.~.

v

(~E"~5)

fb. ~ ~ ~ n(3.w.lt! f~ {di E~~: 2.rwl..ul., N.W.McL~~ trrp~ I

~~ ~

~Ao-I-ht

ka 1>~ B~ ~ t'732.> ~ ~ ~ EAJui_~ 11g1.

-f.Mt ~ 1

(0..)

0..

~ ,,,,~ ~ ~

41\L.........,.

~B~-f~J (b)~~"-t-4~~~ ~~~-"V l (tl.)1 .. Y=o. =o M>,..;..(SO)) Q=IJ b= £Jz./~, C.. =O, ol =2., v:O ~ 'd(~) = ~ 'l 0 (~ ~·12-) .oo '.de~> = A~(~~ ) + B'fo ( ~ ~ ) 1. ~ "fc-x,>= AJ0 (~~) +BYo t~.f.R:). B~ ..tx:t ~ B=o. Cb) Y~.u~·) =o . . Let r.~ k i4 ~.1 J;,Cx) j .C:~e·~ ~lx~) =o ftn. n =1,2, .... ~ ~ ~ ~J ~ (,j ant. ~ Jr;r 2.CJn~ = r. .. > &'?.J

[~c..e-x.)Y'J'+fC.i

{..jn,::

t,/'{ ~n

L~ ~-X.=~, ~ (~"a')'+°f%.'d 0

(n=t,:l, ... )

.~ i.,~2.405', ~ 2 ::::s:s-w, ~3 z

8.C,54)

.it:..

~~~~ ~~ ~~ Y,..ctt-) (J:to.k~ ~+k,

~f~Y),~ Y.ctX.)= A~<2.t1os1'1-X1.t ~ ~c~)=AJ:,(s.s20J1-~ ), ~c~)=A~(S',,s4~,-~)) .a:t..

Su. -pAAt (C) fdt ~' ftn. A= l


(C) > with(plots): > implicitplot({y=BesselJ(0,2.405*sqrt(l-x)),y=-BesselJ(0,2.405*sqrt (1-x)),y=BesselJ(0,5.520*sqrt(l-x)),y=-BesselJ(0,5.520*sqrt(l-x)), y=BesselJ(0,8.654*sqrt(l-x)),y=-BesselJ(0,8.654*sqrt(l-x)) },x=0 .• 1 ,y=-2 •• 2,numpoints=2000);

Section 5.2 CHAPTERS

Section 5.2

58

Section 5.3

59

Section 5.4 10~ Cb) L { S0:t C/;) 3ft--t)tt'C] : L~ 1-t cr.>3:t 1=L f '1 L ~ ~3t] =3 4-: = ~ . (C)

Lf J: :t Ct-1:.) 8 e-3'C t:l.'!:1= q ;t8 * e-3:t1 = L{t 81 Lf e3t 1 s""' O

-

-

t' f FCs) Gts)1 =[' {t2.

11. (C)

8\

s~

s

=+'

I

s+3 ·

~31 = Lfts1 = 2. ii :t"' = ,~±+ ~ f{t)~ (t) =:t'3 1

Section 5.4 1. Cb) I 3x,'+X.='1e~~ 3 [si(s)-1/o~J + X.Cs) = !

= ~ ( __!_ '1

5-2.

-

_J_) ))()

S+t/3

j..2. ~

tit)= k..(e2t_ 7

e-t/3).

i
61

Section 5.4

62

Section 5.4

NOTE: You might wish to discuss the foregoing exercise in class. One important feature is the complication caused by the nonconstant coefficients in the ODE, factors of t causing d/ds's and hence leading to an ODE on X(s) rather than an algebraic one. The differentiations under the integral sign (with respect to s) actually involve the Leibniz rule, which is introduced in Chapter 13. Also, the idea of inverting a transform by expanding it in inverse powers of s and then inverting term by term is important. It is easy to come up with other such examples, such as:

63

Section 5.5 I. (b) f(t) AO

Hct]) -t -:t H -t -ct-l) = e-:t [ rHr.n.. t- c:t-l) = e - e C:t-n =e - e e H<:t-1)

Fcs)

-t

=...!-- - e' Lr ect-t) H
l

_ ...L _ - S+\

2.. . (a..) f l

..... .

:t

-1

tJAAt.

-s--1-

e e

_

S+I -

~'lo)~ f=e~ ~°'-=1 _ e-cs-+1) 1 S+l

f l ()

l

tH(t-2.)

Section 5.6

67

NOTE: For the Maple dsolve command to cope with the Heaviside function (and also with the Dirac delta function introduced in the next section), use the option method=laplace.

T~, ~(f ~C'X. ~co)=oJ,

r

7. (a.)

~t), ~=~);

tk AfJJW'.l

x''- x.::

s2.x- x.

.

.

~ ~ ~ o.1r-rrl. AN\. Ex~ S(b). Hct-t) , xco):: ~'Co) =O . Hct-1) = -trt)

=f- }

%

=+f- cs> 5 -1

~

~±

c..u.

AO

~

. xct) = ~t ~ f(*:) =~r * Hrt-1) :t

-: : s

HC1::-\) ~('t--C) eke

= HPt--•) r~ ~ct-t) tA.i: Clt;J im.~ = Hc.t-1) s~ -'~ct.-t) cit. ~, P". = -Hc:t-1> 's1~x ~ i1""4t ~) H
rtA.r

AO ?U:t):::


s?.x-i=f )

e

=J.Hct-3) S~ (e:t:-zT.+ e-.x)clt 2

A-O

'Xlt)::

Section 5.6

~ Hct-3) [ cs-2t) et+ et-' J.

3

=-tt+t~2t +tH<~-1)[~2.(:t-l)-2Ct-I)) 0.)

x"'' -x:::

bft-1), x
x

F~, s+-1 "'

x"ro) = x "'{o) =o

a

s4-1· s!+t = sL1 - s~+I) -+ {~t - ~t )_, A-
±-

H

= t Hc.t-1)[~
2.. Ca.) J.o,

f-: ~
J~~ ~(-~) ~('C)(~t) =

~ (0)

J_o0 ~(-1:} 8("C)tAt = d(O) ~(O)

~

~

= ~(O) v

Section 5.7 3..

691

NO~:. You ~ght consider discussing this exercise in class. The properties of the delta and Heav1s1de functions that we will use are these:

H'(t) = o(t) or,

H'(t- a)= o(t- a),

and f(t)o(t-a) ={f(a)o(t- a),

o,

(ct)

.if f(a) ~ o if I Ca>= o.

P~: ~~tk ~ ~)== Hct-a.)~c:t-z)

1

'X 11-'X,

= 'Oft-2..)

1

> X.(O)= X. (0)=0.

=

Xlt) =Hrt-2)~Ct-2.) Ao ~ HC-2.) ~C-2.) ::: o ~ Hc-2.)-= o v ~/ft)= H'rt-2.)~ct-2.) + Hc.t-2) ~ct-2) = sc~-2)~C:t-2) + H(:t-2.) ~ct-2)

=~J + Hct-2) ~(t-2.J ~'Co)= Hc-2)~(-2) =O ./ x (t) = H'ct-2.) ~ ct-z) +Hct-2)~ct-2) = sct-2.) ~ct-2) + Hr.t-2)~Ct-2.) = HCt-2..)~Ct-2)

A<>

11

=

A-O

S(t-2)

+ H
~''-~= sct-2)+H<~Ct-2)-~~ct-2)::: ~ct-2.)

v

Cd.)P~: V~ tkt ~ ')'.ft):: 2.+t-.2e-:t+Hct-2)(1-e-ct-2.~) x''+ x,':: 1+ Sct-2)) 'Xto>= o, x/to> =3. xtt) 2. + t -2 et+ Hct-2.) {1- e-rt-2.)) AO 'tio) =1..+0-2 + 0 = 0 v' 'X.'(t}:: J+2.e-r + cSCt-2.) (1-e-(t-2.)) + Hr.t-2.) e(.t-2.) 1+ 2e-t + Hrt-z.) e-ct-~) ___ ) .A.a ~'to)= 1+2+0 =3 v x,11 c:t) -2. et+ srt-2.) e1 t-z) + Hrt-2)(-ect-z)) -2et + gr±-2.) - Krt-z) e-ct-z) 2 Ao ~··+x.'= -~+&ct-2)-l-\r~t- )+1+~t+l-lc~ct--2.) == t+ ~ct-2.) ./

1-

=

1

= = =

(~j \>~: v~ +k ~ i'.J::t}:: 8e:t-4ert + ltio HCt-3) (e 2 H- e:t.-3 )

1-

'X, ~3~ +2 'X. : JOO£ (:t-3) , 1

1

XCO) =4, ~/(O)

t_, _

=0.

=

xc.t):: g et-q e t +too Hrt-3) ( e et. . AO ~ro>
2

3)

=

~~ +~zt_goo HC:.t-3)(z e 2 t-' -e:t-3 ) ~t.-~-z±+200Hft-3)( e'2.:t-'--e:t- 3 ) =ltm~Ct-3)v

Section 5.7

Section 6.2

73

CHAPTER6 Section 6.2 2. (b)

'd': 2.X.~ > 1(0)=0.

=

X0 =0)~=0.z_

=

~ 1 ~o + ff~o 1 ~ 0 )t 0 + 0(0.2)=0 '{)42-=': ~' + ff~1,~· )1l = o+ oco.2.)=0, ~A AO M.. ~M) ¥..t. ~o_j ~ M '(J('X-)=o. (c) aa, ~ (Jo) J ~{X) ::::- 0. (e) ~· = 2.x,e,-1, 'd(l) =-t. ~0 =1, ~ 0 = -1, -k=o.z ~1=~D+ ffXo,~ 0 )h = ~o +1~e-~ 0 h -l + 2.(2.'7l8)(o.2.) 0.0'8"73

=

=

~2.= ~I+ f(?th~I )~ = ~,+2.X, e'd·~ :: 0.0~?3 +2.(1.2.XO·""xo.'2.) = O.S2.'7 ~ 3 = ~2.+ ffXz.,~:z. ).~ = ~2. + .X.2 e-~"ln =OS2'7 +2(r.4)(o.s,o)(o.2.) =O. SSS

=

3. (a.) E~ ~ ).(). ~lX) x.'"+ I. X> o 0.1 o.2.. o.3 o.'f&Ja.n_ ~n. E)(a.,d ~(X)

I

J

1.02..

l.0'1

t.Of

l.04

1.0,

Yao: ~(~LO)- 'j10::

1.12. 1.fc,

o.s-

o.,

1.2.0 1.25

o.~

o.,

J.O

1.30

0.1 l.42

1.5,

t.72.

J."o

f. 3&,

J .4~

f.,4

J. SI

2.00

2.0Q-(.,O: 0.fQ.

4. Cd.) tj':: 'j ~x; 'dcoi =1 -ki. u.....t ~ 'd<_">:: e F~~~

-C(.)X,

1

P~avi:

program sec6_2prob3d

Section 6.2, problem #3(d) Solving the problem y_prime=ysin(x) with y{0)=1, h=0.1

real yold,ynew,xold,h, analytic integer count print *,'Section 6.2, problem #3(d)' pr~nt :·:solving the problem y_prime=ysin(x) with. y(0)=1, h=0.1' print , -------------------------yold=1.0 xold=O.O h=0.1 ynew=O.O count=O analytic=exp( 1-cos(xold)) p~nt :.:x n Eulers Analytical Acc. Trunc. Err.' print , -------------------print '(F5.2, 3X, 12, 3X, F9.6, 3X, F9.6, 7X, F9.6)',xold,count,yold$

------------n

x

Eulers

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0 1 2 3 4 5 6 7 8 9 10

Analytical

Acc. Trunc. Err.

---------------

1.000000 1.000000 1.000000 1.005008 1.009983 1.020133 1.030049 1.045676 1.060489 1.082138 1.101786 1.130226 1.154608 1.190846 1.219802 1.265108 1.298384 1.354312 1.391525 1.459932 1.500527 1.583595

0.000000 0.005008 0.010150 0.015627 0.021650 0.028440 0.036238 0.045306 0.055927 0.068408 0.083069

do count= 1, 10, 1 ynew=yold+h*yold*sin(xold) yold=ynew xold=xold+h analytic=exp(1-cos(xold)) print '(F5.2, 3X, 12, 3X, F9.6, 3X, F9.6, 7X, F9.6)',xold,co$ end do end

E~:>~ ~d ~ o.. f~-~ ~Ao ~ c. f~~ -t4.~)

tJu

'X-

f~ (~=o.s- ~ ~

~~ ~ JJ\Tt.d2. E ~ ~~ ~ E r.J

~~ u~ ~ F~ crl.e.

"'=0.1

l.J302.2<; E'X-ad. '(j at x=o.5 1.10118, E~ ~ oJ: ~= o.s Acc.Trvnc. Error it x= o.s 0.02.94'+

't~

0

~ ~:

~=o.os

C-8'.

~=O.CX'X)l (.13022'

1-t:o.ooJ

-h=o.rros

l.13D2.2k,

1#1302.2.,

I· I !02.2G,

l.lf S-'7~2..

1.12.'730~

1.12~'7'J

t.\2.,,~32.

0.0~34-

0.002.~23

o.ooJ~S"

0.0002.94 't-e

t,c

~cl

~ ~.:.,o.

Dn

~n.:: (1+A-n)

'do

= e, ( + AD~) cc

vt.

>J

t

Section 6.3

4ih.- O"LdJ.n. R-K : Yn+ 1 = Yn

k1

(a.)

= hf (Xn, Yn),

+ ! (k1 + k2) , k2

= hf (xn+l

1= 3000-x.~-~ ~ro)= 2.

2 ncl-o'telttL n=0 !

n

=1:

= Yn + ~ (k1 + 2k2 + 2k3 + k4) k1 = hf(xn,Yn), k2 =hf (xn + ~,Yn + tk1), k3 =hf (xn + ~,Yn + !k2), k4 =hf (xn+I 7 Y~ + k3), Yn+l

7

Yn

+ k1) ·

Exo..cl ~ (,~1 ~)k ~=¥ls-oox.'"+ a.

R-K : -f\. =r;. o2. -k 1;:: {i f (X0 /~0) = 0.02.. ( "31:JOO'){O)i2. =0 *-2. =¥lf(ll'.,, +!,) =o. 02. ( 3000 )(0.02.) 2...z = 0:3 3 ~ 1 =~o + 2. {i,+-k... ) ::: 2+ i-Co +0-3) ~2.15 • ~(0.02.)= .jfwro.02.).,_+ 8 :=.2 .139,7. ~ -= ftfrx.,,"(1 1) =o.02(3oooxo.02.) (2.15') = 0.2.s,,o 1 -kz. ==..ft:f(~1, ~ 1+!1) = D.02.(3000)(0.04)(a.tS'+0.25',,0 )2.::: 0.'/-133S d2.:: ~ 1 + t(-k,+~z.)= 2.\S4--f(0.2.S-~'0+0.'/133S) = 2.q.8,S-. ~(O.Olf)= 2.4'7'712.

fo

4~-er~trt.

R-k':

=*

n=o: -k.1

~\ =o.

Section 6.3

oz

tCX'o/-10) = 0.02. (3000)(0)ii =O ()

75

h 5('X.0+ "''2.>~o+ '*-•12.) =0.02. (3000)(0.01)(2+0) =O.IS2 *3 = n f(X +t/2., ~ 0 +-f!2./2.) =0.02.(3COO){O.Ol ){2+0.075").. =0.13'13S -2.

,t2.::

~q. =~ f

0

(X 1 > ~o -t-l.3): 0.02..(3000 )(o. 01)(2+0.13,35"f2.

'j 1= 'j 0 + tC-kt+2:k2.+z1t3 +k. 4 )

~ l=

=2.14o~s)

~co.02.)

=0. 2."2.1~ =2.1'3,,7

I\ ffXq~'):: 0.02. (30G0)(0.02.)(Z.14-0l5) 2. :::: 0. 2.G,l ~~ ~2. =~ f (~+ ~l:z., '(j1-1--kt12.) = o.o2.{3a:DXo. o!)(z.14-0lS + o.13 lOOf2. = 0.3t1g~c, 2 *..3=1' f(~.+ h/2.,~ 1 +!212.) =o.02(3oooxo.o3)(2. t40l5 +o.17<1-q.st ·= 0.335''8 i4= hf(7(.,_,'~ 1 +~ 3 ) 0.02.(3000)(0.04)(2.l40(S-+0.~35''Bf2.= 0.'3,lt.14-

fl: I:

=

~2.= ~,+t(i,+2*,+2i3+~11-)= 2.41?3'7) 'd(0.04)= 2.'ff]'712

Cb)

"tJ' =40~e1) ~co>= s. &a.ct ~· ri, ~ 1~) ;.o, ~=.D.vi.(2.0x.'"+ l')

2Y\d.-~ R·k':

n=o:

-k=o.02

-k, =~ ffxo;'jc) = 0.02.(40)(0) e3 = 0 ..k.2. ~ -h f( X.1 '.'jo+,k I) : 0.02. (40)(0.02.) e-i3+0) =0. 0007'1 J

';). 1 ::;

n=l: k 1 -=

';lo+ t (-it,+ ,l: 2) = 3. OOOlf-0 • -Rfcx.,~,)

=0.02.('loXo.02.)e-

.k.2.:: hfCXu"j1+k,) = ~:

4th-~

~ (0.02.)

.ooo'tO

=3. 00040

= o.ooo'7~c,

=0.0CXY7""

0.02.(40X0.02.)e< 3 ·00040 +o.ooo1'')

11+t(!1+4_2.)=

3.00040+0.000179,:: 3.00J:2.0J

"j(O.O'f)= 3.0015'}

t<-K: --h=D.02. 3 n=o: J?. 1=~fCX.o/'Ao) 0.02(40)(0)€ ::0 ( ~ +-h/2) "10+ -k.12.) =o.o2.(40)(0.01)e =o.CXX)3'6 J_ Df .1 d .J.. ti -(3+0.0001,,) ~3 - l'\ ( X.o+ l\/2.) jjo + 'tt:.2/2.) - 0.02.\40lo.ol) e =0.0003~8 3 -k.4: ~f(~, J ~ 0 +..E3) :: 0.02.(40)(0.02.) e-(g+.coo ~B) = 0.000'79C, ~' ~o + tc~L+2-kz. +zt 3 +..kq.) = s.0003'8 , ~(0.02.) = 3.C(X)llo

=

=

n=t: ~' =--l{f(x,,~,) =0.02.(tfo)(o.oz.) e- 3 .coo3 ' 6

::

o.ooo7CJ'

,,k2.::: ivf(Xl+"'/2.J ~,+wk, /2.)::: 0.02.(4o)(0.03) e-r3 .C(X'.)3'8+.0003~8) = O.. OOJ l~ ~:s -h f(x,+~h) '':h+*2/2) = 0.02.(qoxo.03) e-c 3 • 0003 ~B+.ooo~ 1 )= 0.0011' "k.4:: ..h_ f ('X.2.> ';11 +~3) :: 0.02. (40)(0.04) e-(3.000 3' 8 +.OOH') 0.00l5'~

=

=

=

~2.:: ~ 1 +t(-t,+2-k2+2.-k.3+.-1<4.)= 3.0003~~+0.0011~1 3.00l5"P} d(O.Olf):: 3.0015~

Section 6.3

2.(d.)

i:::-'d~} ~(o)=I. E)(MXAl...t"....,~ 0';) ~ F~ ca4

fo'1.

~ ~ ~-d\fJ.JJ't l<-k':

real yold,ynew,xold,xnew,h,kl,k2,k3,k4 integer steps,i OPEN(l2,FILE='ques2') h=0.05 steps=20 xold=O.O yold=l.O write(12,*) • n y' do i= 1,steps,1 kl=-h*yold*tan(xold) k2=-h*(yold+kl)*tan(xold+0.5*h) k3=-h*(yold+k2)*tan(xold+0.5*h) k4=-h*(yold+k3)*tan(xold+h) ynew=yold+(k1+2*k2+2*k3+k4)/6 write(12, *) i, ynew yold=ynew xold=xold+h enddo write(12,*) 'Exact solution is' , cos(l.O) end

"· (
n

:>l .OS .1 .15" .2.

.25' ,3

.35 .Lf-

.LJS" .5"

1 2 3 4 5 6 7 8 9 10

1~~) J~ Y

(R-k)

9.987505E-01 9.950083E-01 9.887869E-01 9.801059E-01 9.689912E-01 9.554746E-01 9.395937E-01 9.213923E-01 9.009197E-01 8.782308E-01

Th4 ~:

> wlth(DEtools): > de1 :::dlff(x(t),t)=0.02-0.01 *sqrt(x(t));

del := :t x(t) = .02- .01

Jx(t)

> dsolve({ de1 ,x(O)=O},x(t),type=numerlc,value::array([0,600, 1200,1800,2400,3000, 3600]),abserr=Float(1,-5));

Error,

(in dsolve/numeric/rkf45) cannot evaluate boolean

> dsolve({ de1 ,x(O)=O},x(t),type=numerlc,method=dverk78,value=array([0,600, 120

o,1800,2400,3000,3600]),abserr=Float(1,-5));

Error, (in dsolve/numeric/dverk78) keyword was, abserr, optional keyword must be one of, control, initial, number, output , procedure, start, tolerance, value A dsolve({ de1 ,x(O)=O},x(t),type=numerlc,method=dverk78,value=array([0,600,120 O, 1800,2400,3000,3600])); .

[t x(t)] 0

0

600.

3.313877493498139

1200.

3.852107843603627

1800.

3.967238747167962

2400.

3.992701615124026

3000.

3.998372087416714

3600.

3.999636792316914

H.RJU. (1J\.(. ~ ~ ""~· l~. O'~ fu_ ~ -ro tk ~-~~~~=4.

76/

'd(:t.)::epx..

~ Cx):: Cl) x ( E~)

o. "~'75'0 3 O. ~'S"OOlf 2. o. '88'7'7 J l

o.,soo''" 12..4 O."B~

o. 'S'S33'5 0 .~3,3'72.'T

o

o.~2.to' 1 ~00lf't1 l

o.

o. S7'7582.G,

7. (a.) ~'=Q- 0.01 rfx J

cPt. 0.01.(X-Q

=_cit)

MP ~

Section 6.3

.rx:: u.

15(-te>OQ

t - J..t )

2ao .u.t!u..

M.-1~

771

=-tkt >

=

.2ro r.u-1ao~)+10Q$) ~ =.. Jt)

~.u +20000Q k.c.u.-100Q) -t+ c M-looQ , 2.oofi + 20>()()0(Q ~ ( t/i - f~) = -t-t C ( b) ry'k_ 'X,(O) =0 ~ C = 2~000Q 1"" ( 100~) • Q..k,) lit Ml.,\, '™- ~ ~ ( lfX .. 100 lQ) GUV ~(lroQ-,/X) ~ fdM/.u.= ~l.ul MM~ 'le~ -t4 iMc )>A~

·~I I'~. 3~ ~o):OJ >M-~ ~6 =4~~;~, l~-R~..k >O, ~~~~I I ~~~. rp~, ~Q=o.02, 2.0ot{i. + 400 ~(2.-~) = -:t + i+ao k~. rp~

}'}\~ ~AMc( . f~ (2.DO;t~"(l/2.)+lfOOf k(2.- ~/'(1/2.}) = -Gl>O +4eiPJ;k{2.), ~)) fdl. 'XUOO) > 3.313'8'7'744':3. ~r:,~J.:70 120D;Jgan} •• ., 3'® X(12D?>) = 3.a5"2107&lflf

t"'-

rJ

X.(l800)

=

X,(2.lflro)= ~(300())

~

3.~,72.38'7~'7

3.,~2.70) ' ' 5

= 3.~CJ8372087

'X(3b00)= 3.~~~'3'1~2.

~~ ~'?8 ~) ~ E't.~'ra.),~~)

tv(

At.t.~t4 ~

~;t~ IO~oJ.~: . 8. En""' Ctf Ao.~~~~ .1' ?~-?t~o)~:to c.. To~U ~ .1.t C=l ~ r=2.. ~) fo't ~ rn~... '°" -It M- ~ . -h= 0.1 4'= o.of ~:; 0.001 Cf,t =i:k-z. = 0.01 o.OCX>l o.oooooJ +-- ~ H~ c, o.5Cf!::: o.s~-i.= o.oos o.rxxns o.oooooos +- ~ 4 ~f, C-h'l.P== i.h = 0.0001 o.ooooooot o.ooooooooax>f-t-M~~ ~. W.e. ~ ~. ~pAl'Nl. J-[~n.+Ol-ltJ ~(1Cn)+~frx.n,~CXn)J-k] ~ "'· C...U. it" F<-lt). ~) Ji.a~~~. F'<-h) = ~ + ~ ~ = o<.# + ~f[x.,~c?G.l) ~

i

*

M

o(fx + ~f f'cl Cl~) = '(j+ fa.f +hCf+Cclf~+~ff':t)t+···J} ~ = 'j + (a.+b)fh + (olfx + ~ff't)b-h-z. + ··· v

RHS 1

=:

Section 6.3 10. (a.)

#include #include FILE *fpt; main {)

/* Defining the variables now */ char c; int i, steps; float xold, yold, xnew, ynew, h, x_n, y_n, kl, k2,k3,k4; fpt=fopen("answer","a"); h=0.05; xold=0.0; yold=l.O; steps=20; fprintf{fpt, "\n"); /* Starting the loop now */ for {i=O;i< (steps);i++) {

kl=h*(yold+2*xold-xold*xold); k2=h*((yold+0.5*kl)+2*(xold+0.5*h)-(xold+0.5*h)*(xold+0.5*h)); k3=h*((yold+0.5*k2)+2*(xold+0.5*h)-(xold+0.5*h)*(xold+0.5*h)); k4=h*({yold+k3)+2*(xold+h)-(xold+h)*(xold+h)); ynew= yold+(k1+2*k2+2*k3+k4)/6.0; xold=xold+h; yold=ynew; /* Writing out the results to the file now */ fprintf(fpt,"The solution y(l) using h=0.05 is %2.9f \n", ynew); fprintf(fpt,"The exact solution y(l) is %2.9f \n",l+exp(l)); fclose(fpt); /* Result obtained- closing the file */

The solution y(l) using h=0.02 is 3.718281819 The exact solution y{l) is 3.718281828 y(exact)-y{estirnate)= 0.000000009 The solution y{l) using h=0.05 is 3.718281474 The exact solution y{l) is 3.718281828 y(exact)-y{estirnate)= 0.000000354 Thus, computed order of the method from Eqn.

(b)

<'.5MX'~:

The output after {x_n+h/2) was changed to x_n ..... The solution y{l) using h=0.05 is 3.701259429 The exact solution y{l) is 3.718281828 y{exact)-y(estimate)= 0.017022400 The solution y(l) using h=0.02 is 3.711558039 The exact solution y{l) is 3.718281828 y(exact)-y{estimate)= 0.006723790 The new order of the method -

!..:..Q.!

It is no longer a fourth order method.

{28)~

4.0075

781

S~ ~ ~

-fa't A,B, c ~

A= (2X-~fn +2~! f~+,-zrx~ -fn+Yz.. + 3"n.f"'"h + ~t\fn. . 1 ~-4Xn.fn+'t:z.~+ f..,,-#t)/~~

B: -(4X.n.-fn+4Xy,,fn+1-8~fY\.f.Va.+ -'fn-k+ Tn+t'lt-4fn+h~)/1t C:: 2(fn-2.fn+Y2. + fn+•)

Section ().4 1.

Cb) '})'=4~) ~(2)=51 ~''=4-:r.'= - LJ.tj. ~ "j= A~2~+ (Scp2~. t' =-'d, ~(2):0 '1T'~ 1! = 'd'l4 = (J1ct:>2x- B~2~)/2. Tk "d(2)=5'= C~4)A+C~4)B } ~A= sA-\M4

=

~a 'i:(2)== o (~'t) A- (~'t-) B 2. 2.

•

Ao (I)(~~

A.O.

B= scp4

'd(X):: 5(~4~2.%+ e<,>4C(:>2.~)

=5C'.(.:)(2X-4) =

~(~) =f(~4~2.'X.- C<.>4 ~'-.'X..) -~ ~(2.X.-4) H~, .ll](a..c::t. 'dto.2) 4.'0530427 thwl rico.2.) = -o.,~354SBS,. ~: ~I= ~ 0 -t-f(XoJ~o1~0)-h "(1o+ ll2ot:: S-+0 5 ~,= ~o + ~< ·· )-h := ~o -.~ 0 .ft o-S(o.ri.) _:!

=

=

~ ~

2.ntA.-(5'Jt.J..u-t R-K:

=

=

).QI

= 1n+• = ';JV\+ ~ (--k,+k2-) +t ~Xru';)~) >12. =-h f(%V\+I >'dW\+i,)

-k1 :: Ao ftn.~ ~ 'd'= :f{x,d)~)) 'i!= a(.x,j,:) .At°~ 'dn+1 ~n + t cl, +!2.)

= ~n+I =~n + 1: (.Q, -+ ;,2.)

~f(Xn1'd~,'Z:r\), _.k2. =~f(XMI > ~n+~11 ~r\+.Q,) 1,::: ~~('Xn,~n,%:'1.), 12.:: -h~(Xn+L>'dn+~,, '2r\+~1) In~~~J fortn=o, = -h(4~ 0 ) C0.2)(1.fXo)::o ~. :: hC-Jjo) co.2)(-5) =-I i:,:: -h,(4)(~0+.0.,) (D.2.X4Xo-I) -o.B 4 =-/t(-('()0-t-.k, )) = (0.2.)(-(5+0)) -r 'd• =~ 0+ t(.k,+,k.2.) s+-£ (o-o. S) 4.'1 (~ ~!) ~.:: 'r:o+ ± {.el+.Q2.) = o+ f (-1-l) =-=l )(,:

*•

= =

=

= =

=

4·ih t5~ R-K: ~ ~ ~ ~ ~ -I!,= ~HXo,~o,lo) =(O.'l.)4~ :: 0 Q1 :: t~

=

fi,):

( " ) e (0.2)(-~ 0 ) =-I

-k:z. =-hf (~+~2.J ~o+-kv2., ~0+Ja12.) = (0.2)4(~~ .Q1/2.):: -0.4

) :: (0.2.)[-(~0+i1/:z.}]= -I 'k 3= ~ f ( ~0+~z, tj +-k2/2;e +.l.:z./2.) = (0.'2.)4(l: +Qz/2)= -o.q.

~z.::

-h 'C

"

0

0

0

l3 = t~ ( =(0.'2.')[-(~0+!2./2)]~ kq_ =-hf (Xa> ~o+-k3, i:I) +.!3) ={o.i)lf-(i:-0~~3) =-o:u,s .lq. =-h ~ ( '' ) =(0.2.)[-(~ 0 +.f:3)] =-0. <}2. ~I ::: "'jo + (-k,+2:k2. +2:k3 + ·t4')/, = 'f.,05'333:33 ti

~.

~(x) = 5'Cd:>(2x.-t.1-) c(X)

-o.,,

= 2:0 + (~1+2.l:z. +2~3+14)/, =-0.9133333

3._(b) E~:

)

=

x =3

~=5

~=10

-2.ogo134183 4. 8008.5"1433 -4.7882,'7tl02.

=-~~{2X-IJ.) = -2..2132.'-l-35'1

0.,,8538'74~

o.7J"J'7S82'2.

~: ~~lk~RKF45"~~)~~~

81

Section 6.4 ~(:DE~):

*

~ (f~ (~C~) > X) =4 "i:.('X,,], ~ {?::(~), x,) = - ~(?G )) ~ (2): S:, ~=~J ~=~([3,5,IOJ))j

r

~ ~

c~, '(1(~,, %;(?(.)]

=

-.2..080'7.342.IO

3

s- 4. 800851531

[

4. Ca.)

~'Ct)= F(t,~ 1 ~ 1 i:)

= G( i!'{t} = H(

~ict)

Eu.k:

-Xn-+l

JO

='d- t

;

l

0.71~7583300

'X-(0)=-3

)

.,

o. '"~ 85381'4~

-4./882.,7,bl

= i! ; ) =t+~+ 3(~-~+I) j

••

-2..2.'732.'4-3S'77

%:(2.): 0] >

0 r,{O):: 2...

"j(O):::

=~n+F(tn,~n/~Y\,~V\)h

~n+1:: ~n+ Ge

t:f\+, = ~n + Hl

)-h

"

) -h ~I = k f C:tn> ?lYl>~n >~Vl) .Q• = ~ G( " )

2nd.~~ RK:

~He

m, ::

11

)

1<2. = ~ F(±"n+I) x.n-r*-1> ~n t .Q, = -h G< .. m2..=

4-thO-.nJ.ut RK: -k,

hHC

.e., ~n+ mI)

..

)

)

~n+I : X, Y\ + f (--ki + ~ 2. J ~ n-tl = ~ n + f (!a + ~:z. ) rn+1= ~n+ t(~,+Yri-z.) = tfCtn, ~t\,~~,r.n)

.e.,=nGc . ) rn, =hH ( ) *z. = ~ f (tn+~k.J X.n+ *dz>~~+ .Q,/z, ~n+ m,/2. ) ~z. =R_G( " )

mi =~H c " ) -k.3 :: { F( tVt.+ 'f../z, Xn+ *-2/2. 1 ~n + l.dz, ~~+ Y't\1/Z) 13 =hG( " )

m3=ftHC -k.4 = ~ F ( t'n+l ~<1>

= ~G( m"' = ~H (

"

> 'X,Y\ +

-k.3 , ~ n+ .Q3 ' ~ V\ + ~ 3 ) "

)

"

)

~n+l

= ?ln+ 1;(..k,+2.~z.+2.-k3+!4)

"J>'\+l

=

Z;n+1

= z., + tcm,+2.m'2. +2.m3+W\4)

'dn + 1;( .ka +2~z. + ~3+ .Q4)

)

82

f ~(X), i:lX.)]

Section 6.4 ~~MM,{.\. M/'~ ~

Sk1. ~--

Q.NI.

C5"~ RK ~' ~~
:to

83

~ x,>"(j,,~ 1

~ X2,~2.,r2. fdl ~ ~ .VX~· E..u.k: 'X.t = 'Xo+ f(.t0 ,'X,01 ';) 0 ,~ 0 )-h = X.0 + (Aj 0-I)~ -3+ (0-1)(0.3) 3.3 ~I =~o +Ge .. >""- = ~o + ~o~ o+ 2.(0.3) = 0.b '"i, =~o + H( " )t =~ 0 + [t0 +1'X1,~ 1 ,~ 1 )-h = ~ 1 + (~ 1 -l)h = -3.3-t(O."-l){0.3) -3.42. ~2.:: "jt -t (;-{ '' )-h.: "Jl + ~I -R :: 0." + (3. 8:(0. °3) = J.'74 ~i. = c 1+H< )"1. =~.+ ct 1 +~ 1 +3(r.- 1 -"j 1+1)}h =3.8 + ro.3-3.3 + 3(3.8-0.'1+ l )J(0.3) 8

=-

=

=

=

='·'

2M O'Nlvt RK:

= "'-<~,,-1) = (0.3)(0-1):::: -0 .. 3 =-h ~ = (o. 3)(2.) = O.b

-1!1

.e,

=

= -h[t 0 +'Xt0 +3(~ 0-~ 0 +1)] ::(0.3)[0-3+3(2-0+l)] LS --ki. = +t.(t;l 0+i1 •l) = (0.3)(0+ 0.,-1) =-0.12. ~t. = t ('%:0 +M,) =: (0.3)(2.+l.8) = J.14 m, = ~ (t 1+ ~ 0+-1<: 1 + 3(r:0+m1-"j0-11 +I)] = (0.3)[0.3-3-0.3+3(2.+l.8-0-0.b+ t )] = 2..88 X.:: ?Go+ -£C~,+-k2..) -3 + i(-0.3-0.rz) = -3.2.l "(11 ::: 'do+ t (1., +..Qz.) :: 0 + i (o.,+ l.J4) o. 87 7! 1 '2:0 + t Cm 1+ VY\2.) = 2 + 4: (I. 8 + 2.88) = 4.34 M1

=

=

-k,::

-kC"j 1-J)= o.~(o.e1-1)

=

= -0.03"

.2., = ti.~. :: 0.3(4.3lf-) = 1.. 302. m1 = -h[t 1+X1+3("! 1-tj,+1)]::: 0.3[0.3-3.2J+3{4.3'f-0.87+1)] = 3.IS ~i.= h("J 1+.Q,-1) := o.3(0.8'7+1.302..-1)= 0."35H, ~z. : ti_ ( ~. + M,) = 0. '3 (4. 3tf- -t 3.15") : 2. 2.'f'7 >Ylz.: ~(:tt+X 1 +i 1 +3(l 1 +M1-~ 1 -J,+1)] = 0,3 [ o., -3.2.I - 0.03 '} + 3 ('f:3't +3.IS -0.8'1-1- 302. + 1)] = 4. S~IS"

t

= =

·i

~2.::: X 1 + (-k,+.k.2.) -3.21 + (-o.03,+0. 3Sl') "<''Z.:: ~· + ~ c~ 1 +1.2.) o.87 + f ( 1.302.+ 2.2ct1) 'lz_::: r.,+ tlm,+mz.)= 4-34-+f-(3.IS"+lf.6'15")

= - 3.05'3?

= 2.,445 =

8.%075

1J.t~)t~ fo ~Ml~~~~~ 1 WM~ rJoMJ~~· rpfu ~ Cc1WvW\~ . ~ 3

~ 0 ..

( f cL-H ('r-(t)) t) =~( t)-1) d4l,( ~(t), t) = ~(t)) ~(Tit) It)::: t+ 'X.(t) +

(c(t)-~(t)+ 1),

'X.Co)= -3, 1fo)

=o, ~to):: 2.}, f ut>)~lt) >ett)]),;

.tzet - 3-.tt ~ ~(t 1 ): X.(0.3)::: -3.\'185"12.'70'1 J 'X.(:t2.) = 'X.(O.b) ";)<:t) = C:t :2.t) e =-2.~'f'Ll-03'1232) 'jct,)= ~to.3) =o.,314025'17) 'l:(t) = (± +4t+ 2.) et 'j(i:z.) = .Ai)fO.{,) 2. t34250S32i;l,

'X.(t):

~

I

=

~ct,)= lf.4410354?'1) ~rt,.)=

8.ta132. 8.5'+~ ~zna.dl~RK~ ~ ~ ~, W J ~ /t'-O.vvu11.P~ ~>~~~=o.3~ .teo ~

Section 6.4 ~(~ ~=u

; l~=~

.tt'= -~u- ~cl+ -[E'lt)

ce) (~)

~' =A..t.

,

.AJ..'

;

L~

.u. (-2.) =4 l\J"(-2)

;

=0

2. .U.':: -2.X + 3"(1 + lOCd'.)3.t' ; M(O)= -I

'd':: (\) f\f

~. (b)

j At(O):

~(-2) = 7

= rv '= 2.~.u. +3X; tt,' =).;{_ (\j

84

1

-:::

X-5"~

'X..(O')::

;

~(0) =Lt

j

~{O)=

3

~'::: -.3x,~ + '62. j 'd(o)= I ':l. 1 : Lt j ~(D) =2.u,'

=ru-

nr'="j2...M.-T:..

)

WO)= 2..

j

ru
'P~tk~~~ ~{DE~):

&4k (f ~ (~C~),X) =-~*X.~ %:£~) + XA2., ~(~l'X,),~ )= M.(:t)) ~ (.U.(~)J ~)

={\j{X),

~(ff(X,),~): "'j(rx,)"21 U{X,)-~l~), ~to>.=t, %:(0)= 2.J

.uro)=2, ArfoJ=-1}, f ~ex.~\ ~Cix.), M<~),.Are?G)]) ~ = ~) ~

= ~.((1,2.])); ~~~~: x,, ~('X.), 'Z;(X), .U.(X)J IU(-X.) -3.J 83Gt 72.0lG, 3.2.~/05"3188 . 25515',3258

I

S.052.2.Jl~38 -2~.027334LIO -2.47. 8533'142 -12.SC,.0'70148

2.

1wkl14 '7.

- 2.5553"7435'7

~~ ~ ~ t4 ~~fort.

~(DE~):

.

~( f ~(Ycx),'X.,) = Uc'X.), ~(UC~),~)=Vcx), ~(V(~),x.) ='X"'2.•Yc'X)} Yeo>= OJ Ufo)= o, Viol =t}, f)'c"X->, Ucic.>, Vcic.>~, =~, ~=~(cs, 1, 2.J))j (~ '/, u,v tt1\t. Y3, U3, V3) ~ x. = O.?, I, 2. ~('t) =.12.'501~~, .502382.75"4) 2..3\22082.3

11ft

Ne4., ~(f ~~CY,x.)=Uc~>, ~( Uc~>,'X,) =Vt~), t9{C:;<'X.,, ~)= x"2*Ytx)-x,/\4J Yro>=O, L!Co)=O!Vrw=o}, ['tc?a, Uc:x.),Vc~~1,~= ~) ~= ~([.s-,1,z.]))j (~~ ~ Y,U,V tlJl{. Yr,Ur)/r) ~ "I..= o.s 1 2. . 'trlt)~ -.rxxtJ3?2.0 -.004?,SSJ -.t;,z~J'lflt F~, (197) 'd
r

"a(J)

= (2..0){.5"0238275~)- .00lf-1C,5~l = 1.00000000

.k.l.u.J, ~ IAM¥;J_ ~- ~ ~-°R:> le.~> ~J

tk~&~~alrnt ~~~4<~~.

";j(X):-x_2.,

~

Section 6.5

8 . (t!AJ)

=

~''- 2v.i + ~ :: 3 ~x; 'd(o)=r, (j {2.) = 3. ~Cx) = C,Y.cx,)+ Ca. %Cx) + Yp("-) , ~

LrY.J =o: Y.'= .u.

; Yato>=1

J.,t' = 2.'X.Lt.-

Yi=

L[ Y2.] =o:

Y, ;

(\j

N"'= 2.'X.r-r-Yz.

85

.~(1) ~

=

'f,'Co) =.UlO) 0

;

Y2 to)= o

;

Y.jJO)=fffO)= I

"Yr=.w-

L[Yr]=3~$(.: j Yr(o)=4t w'=a:xw-Yp+3~'6; Y/ro>=wto)=h. rr~, ~(O)=i = C,Y;co)+ C'l,'1.ltJ)+ YpfO)

rp~)M.t.v.Y~ tt=i,~~ c,=o~ J.J.d. cto~ _If. ~ 1-0 ~ Y. t'X.). d(t..)= 3 = ~. 1. (2.) + c'L y2.(2.)-t 'tr(2.) = c2ic2.)+Yr<2) ~ C2.::[3-"'{p<2>)/~('2). ~~h=o)~·

= c,+a..

MC't)

()

= 3-'frc2) '!, C'X) +Yi: (x,) . °%,lz.)

Jj r..t JWk. ~('!-)

f

'Z-

*

iJ: X= O.S, I, 1.5, ~ 1 (a.ei; ... Q~, ~ -x.=I~ ~ -(.:n ~) ~

J.M(.~m~~ ~(:DE~):

~

( f '91(Yrx),'X. )= f\f(~), ~ (Nt~), x)= 'l.'1. 'X..f. nrr~)- Yr'X-') 'Ito>= o, N"(O)= I},

~ ['.C:;1:),r.rf'X.>], ~= ~, ~ =~ ([o, o.5, 1.0,1.s, 2.o]))j

r ~

,

1.0

2..o

t.S

0 , 0$222.0~tf52., 1.2\,JC,ll«BO) 2.,Z37,23?2., '7.G,3~1~012.I

Y2t'X)=

~~

O.S

O ,

4X- ::

(Yio Y, i.w)

(rcq.{C Yr~1, 'X.) =

J.Ar(X,])

~ ( Mf(X), 'x,) = 2* 'X.._• >JS"(~)- Yrx) + .3• ~('X..))

=

Yro) =1~ wto)~o JJ f YtX-), .wYr) ~ ~ x= o) o.15 , 1.0 , 1.5' , 2.0

Yyr,,..)-= o,

o.~325J354BO, o.,,~,13008) 1.'733,75'c,2c,)

'd<~) = -o.4131403% ~c11w) + 1rr't)

'I~* ~

,()()

~

=

0

' 0. 5'"

1l'X.)= I (~), 0.'71''7'814 l

Section 6.5

,.15',03?371.

' 1.0 J

) 1. 5

) 2..0

O.Lf'2.00'5''7, o.e,4~~g1~ J 3 ( ~)

Section 6.5

86

Section 6.5

88

Section 6.5

89

~n= c1+c2.c-2t+ ~(nt.-in+2)· 1{,.(b) ~(~(n+1)-2~~(n)=3¥~(n),~(~))j ~ .. n ~ M(O) 2~+ 3 4/)w\{n-Qcpl -~(n-1)-2.~(r\-2.) - 3 ~I 2. 0

""on = ()

-

-S+4ct:>I

To~~~~~

~n=

[ '()fO),

3~']2."+

~4':>1

s-,

cl)~

IV1.

~ > ~) ~ (1+~2.)/2

3 _ 44=>1 5

-S-+4C<->I

1s
J1t.-~ .J:"a..a,

2~2.

(4~5fZ'i.(-.~n-~n~1+~1C!(.)n _

2 ~ne<.>2. + 2.~2. cnn.J

:: C,2.n+ 3 ((2+~-~1-2.~2.)~n+ (~+~1-~2.)~nJ 4er.:>1-6

v

.

Section 7.2 CHAPTER 7

Section 7.2

90

Section 7.2

0

I.----.---'----'----'--'-'--'-'----'-'--'--'-----'-'-'.......

0

A

B,C

B,C

91

Section 7 .2

ce) ~. (d) 'X}:~

]~:-JE

~': -2'X,} 6J

Ao

~

'd?.-+x"'= c. 6N
~ .o.o- c~~(~.e.,fdt~d-~

~))~~"'.JC n
aA

C~cO.

Cb)~( ["j>-2*x"3J,C.t,lll,1j], t:::.o.• w, [[ors,o].[0.,1,oi (0,210], [o}t·o] ~t-= -~

JJ

~= c~}~1, ~=THtN);

92

Section 7.3

Section 7.3

93

Section 7.3

Clf +.r) -f "'"'r) ~

~

(" =~(nrr/4),.+I

(qf-tf>~-(')) ~ n:: OJ :t I)± 2.) •••

2. YJAI. fo'l. ~·; x'= x~+'J.,_ ~ "O'= xi._,.'J,..+I ~ ~. 3. c} (t>-~)X-bY•o ?~ [ ~D-Gl){D-d.)-bc])'=o J>-a.1 -cX+ CD-d.)Y=o) .J- ~ [ " ] X=o. ~' Y''-c1.+J..)Y'+ ca.J..-k>c)'/= o ~ :: e>-t · ,\'l.-Ca.-rtt)A ;-c~-bc.)=o) )\= a.+ct ± ca.+a -i._ *~-be)

=[0v+d.i({a.-d.)'L-t4bc]/2.. v' To~ C3,Cq, ~ ~ 1 c Cz.) rt {Jo).,.,,Jo te;): p~ ~(,a.) 11

~ . >. 1C, e»•t+t.:i.Cz e>-~t :a.Ci e.\,t +a.ti e>-at . +DC3 e)\1t' +hC11-e>-2t ~ ~ eA,t ~ e>iit AN. LI) J:" ~ ~ A,C,:: a.C,+.bC3 ~ C3 >.,·i/·)C, Ai.Cr.= o:Cz.+hC11 C*=(~a.)Cz.. P~Clo).WO (.,b)......,,..U ~ ~~.

=(

~ €)~ ~~ S ~~~. 5. (b) lit ~«EC J ~ol: S. (5".I) ~ (lfo) ~

4. No.

rotiOJ

~C-~'s::: ~(l(°c-;s)+!(Xs+~c)

x's+~'c =-fcxc-~s)-ll(xs+~c).

E~J:.M,

.

u

3

X'= (flci.-t-~c:-J8si.)x_+ (4c~1?cs*11si.-_fgc~)~ ~'= (-.Ii SC-4s;-Hc --Ii sc)x +(--11<:St,/is;+11cs:JBci)~.

To

~

;d': ~1-~ ~~I: -({~X,) Ait ASc'Z.+'lcs-ucs-.msi. =o e11..) t./8 (c1.-si.}-ics-=o >~ ~ -'lcS+ tJ8 Si.+ ucs-fsc."":o rlL) ,f8 (s:.c") +?CS= o v~ sic: ~=t.) ~ {1-tt )- i"t= 0 MJ t ='If§ At:) Of.= r;.;:;}(1/,Ji )) r..,;.! (-,[8) ),.41°, -10. 5"3;

~ ~~ i Mt . ~: ''·'ff x

= ~

,-rs

~

~ x

94

Section 7 .3

F~)~tv{W~~~~1ts ~ (/ N.tt cJ-0 ~la~ +kt r;~= (q.c-z..-.[Scs+usi--,fgcs)/3 ~ ?!~= (4cs-~8 si.-11cs-t ,Ji c'Z')/3 ll1\t. >o. V.&J.) .t =fg ~ 4-Y[t+11t.t.=4-2,m[Y.8 +11/8: I..; o 3/ci8/9)

f':

OMJ.

'tz.= 21fi~j~;l;1.-tll:: 4V

-Ji~ (l?. >o c OJNl. ?/ >0.

~,::t:

1

~.

.Ci:: Cz+Ac. +ACz.t ~A ~r-to/J) ~NA~. 'XCa+ Ci.t fJ

7, fJ'tlw\. ...e..1w- {19,;..)J >.1~0 ~ 1\2.-+ -ca ~ t:~co) Ao

"J "

~f>~~AA~: 8. ~, ~ co.~-'X. S~~~ ~~)...

fdi~.~JNf;~~W(

~~~~~ ~;.-r~ ci.t ~ ~ ~ ~~·~. 9. UAC.. ~ =( tti- cA ! 1-(4-..-tA.)...... £ +_lf_l>_C 1I2. . (a.) A= (2 +,m)/2 =3,-1-+ ~J,JP.J.... s~

'cl =l
'X/= 'X.+J
=

K=2

1-+ 1-t-K=4+K

K1 + I< :1<4+ K

)( =±2..

O'tn, K=+2J ~'=3X-+~~' ~ A K=-2> ~'=-x~~ ~· (cL) }\, =(-2 ±10+12.)/2. =-1 :t{! -+~.$~~ ~:::~XJ ~'=-~.,..31
K=+'/1/3

(wJa.Jk),-Y"3 (~).

K=l/,£3

x

1<=-1/~

95

Section 7.3

ca) )\: ('l-:t 10+~ )12. =3,'

.

~~~.s~~

'd=KX~ ~'= 2'X-+KX-

=X,+2J<:t

} 2.+K:: 1+2.K

7

J<: 1<=-+1 ~A=3J K=-1 rco>..=1 C~) A: {-,1:.flf)/2' -1,-4 ~~tNzlt.. ~=l'X ~ K't.'

=

x'=-3~+Kxi,-3+~= r-~t< I<~'= ~-3K~J 1<:±1 1<:+1~ >. =-2, 1<=-1 ~~=-4

NOTE:

x, K=-1

x:::-lf-

HcJW"' ~ W< ~#.tfht ''~#JI~ NE·~

SW(~~~~)~(,>)~ N\tJk, SE~tj)'?

.k.c~>, x =Ae fir -t Bet tM-J. d::: x.'-2x =Aef.t_s et:, Ml ~""x «.<> t~+o0,~ ~ ?O =Ae-2 t+Be-41-r a...-J.. ~:. ?l'H:i =Ae-zt -8e-4~ .oo ~"' - x, ~ t_,, -o0 (a....J.. ~"' ~ a_o, .t .~ + oO) .

10. '(j:~x,

'r

x.'::A..~+KbXJ i=~~':cx:tKcl~J ~ a.+hK.:: (c+&.1<)/K ~ i~ 11,N\· "'- k,~ ~

~ OJI, c'?,2. ~ ~. I I. ~ A.::: [A.+c:l ±~(a,-d.)i.+4,l,c. ]/2.. (Q.) >..= (2.±-lo-1c. )/2. J ±ii -->- ~ ~. -=(S-±{t+tf-)/L=(5!~)/2.-+ ~~. 12. L~ MO~ (A),(~))~(~). (~)~ (d.)mt 4 (~

=

=

~1~~~#..r~~aU i4 ~~ ~, Jnj- fo'1- (~) >Ht ~:to ~ dk

~~~: '::KX~

~'=2~-K'X.

K~'=-~+3k~

1 -+

.2-t<=

-:,14'~1<.) k

K=.,18034
96

Section 7 .3

C«-> H~rk~ ~

t"4.

~1~~? FnfW\. X' 'X.1- 3 d, ~'

=

WC

.0.U.

thit fo'l )(: 0

~ ~ >O, Lvt -k %' ~OJ p,o .~ ~

~·

97

Section 7 .4 Section 7.4

(1)0): ~': O(X·l)t~

1"'-=

~'=-t1-cix~1)+0~

'4.:0,b~l,,

c: -'I-, d.:O)..«>

(o±.1~)/z. =± 2i..,. ~

O't~, .fn,f

~ /~if:_~-x.'t-x,"·: I> --+-\~~--+-_..._I--+-+-___..

~~I

M.~~~k a..~.

98

Section 7 .4 (b)

~=(0±~0+8.. )/2 ~'::-I (X-1) + O{~ -1)) ~ ,4AJJ1.c..,

x

(1,1):

(1,-1): (c.) (-J,O):

c1,o):

1 :

O
1

x'= o
A=(o±,Jo-s )/2.

~':: -l{'X.•I )+O(~+-l) ~~di~ x':o{X.,.I)+ 1(~)1 )\:(0±~0+4 )/z. I ('X,·tl) + O(AJ) _,, ~J&

'() '=

x'= orx-1)+ 1(~)

{)'.::-J(~-l)f-Ol~)

1)\ =(otAo-q. )/a.

-+~en~ aOw. o it" ....JJ. k ~ ~ £...,....o.t. i1't.. ~~ '4.~ ~ 'X- 11 :(1-x'-)/ti+x,,))

O,,

~~~. rct> s~·F Al" =nn-1.i cn=o,:1:1)±2, ...) x = (~-tlTT/a.)-(~-nrr/a.) 1

"='

'j' = epnrr ("'-ntt/2.)+ ~nrr (11-nrrla..) AO ~: I, h =-I, c: tl. =cr:>nTr = (-l)"-J ~ A.: (I+ C-1)"' .:t ~ (1-C-1 )" )'- -4(.. t )" )/2.. · n~=> >.:: 1±i ~ ~~~

no-M~~=:!."2--P~ (e) Co,o): x'=oxt•~J A::(-2±1-4---4)/2=-J,-1

".:)'=-~-2, Tht.~·~no-tt ~. ffiJ: pi.-'+i=o:~~.SH~,

r.= a.+~= -2~i=act-Joe=1,

~ ~~dl.~~~dl ~~~. (-1,1/2.): ~'= l(u1)+or_,· 112H A=(-2±ff)2. 'j'.::-(~+1)-2(,-1'2.) J ~ ~ (1)~2)! ?G':: + ('X--t)+ O('d+V2)J "-= (·1!.ff )/2.

':J'= -(~1)-2(~+1/2)

/)I()

_,.,, ~

<1)
99

Section 7 .4

100

Section 7.4

101

Section 7.4

x' =<~+,) + o(~-•1zJ

1j'==-c~+1) ... M ~

1

t11., "X'~

X+oY ~/,.,(.Mil<. Y=·t<.'X

2(~- 1i2.)

"'f'=-X-2Y, ~ ~) X'=
13..X'~ ~ ht.4- ~ ~. ~ ~ ~ i!O~ ih.~ {~=-V2).--(l...i"N.~r;,...~ ~J /#(} ~ ~ .f.,:J. 4' ~ Y= KX-~ IN<~ ~ ~ X=KY: ~ KY'= l
?r:O

(b) ~

!'J

(G) ~ Cd.) h\-0

S: (o.)d-tb): Su F?.S-~'1. Ce). Jt~ ':2.~

Let'--o~m.:

.ht~~ n::s~~:l"k~ wJJ.k~l011'.:t ~r'1·"' M~~ ~~ ~O,O)J2!f',OJi('f11',0), ••• ~'~ A<>

~~M>~~~~-~ik~ ... ~ ~ (lr.o)~ {Jrr,o) . .Dt:u~..q:t.,~ m,o). ~~

.~AA(,

·x v~y':l.t\-3t. v v di., X :0 +11,

1

'X., :~

~ O(~Tr)+I~

~'=-lM-Avv\x= l(X.-Tl')-3M Cl () t1' Y:kY.. 'a~ X'= J
-2

..

102

()J(J., -

3 :302.8 . So~

rr~oz,

~

~~~ Ul~ •.__!p

S;-··-~-.. --- fdt ~ ~ 0-t .@ ~r. 0 ,~J:~-

·°"

1T-:02;.°°'1

1T

Section 7 .4 rr+.02,.~

~ 11'+.02,-.oG."

~)J~ ~(C~t3"~-~Cx)],[t,x,~J) t =-35•• 3S-) f [o, 3.J,,S--,,.oo, J ,[O, 3.1'1~)-.o'' J)

[o,3.121S~,-.00'1J, (0,3.12.JS~,.O,,J., CO,~.lf"t478.,

(O/~.m7SJ

.00, J,

-·°'"J, ro, ~.~78,-.00" ] [O,~,tfOcf-'78) ·°''J]J 1

~:.OS-> ~=-2 •• 14)~:-3 .. 3J~= [~,~J)j ~ J ~ (Tr\- S ~ ~ -~'to uf.JQ_ M.. »

ikw.

~>.o~~.

~ Jt::.3,,~ (32.) ~ S+=Cx.+.'!J+)=(3).'3) W S_ =(%.. ,~-)= (1/3) .1). E~~ ~ ~ (31) ~ St: x'= -Jl.{'t.-X..:t)+ (~-~:t)

6.(a.)

~' =(2 x.;_ -z.. (x-x.,. )- (':J-'d :±) l+:t:.t)

ar s+ ~ ~

-

x:= -:sx +Y} A.=c-1.~±1.
:::-.221 -1.08) ~.NI,()

IL~~~.

s~~-~~~iis-s+:

Y=1c:X;

X'=-.3X+ t
x:

"'=

S-LA-k ~ ~ ~~ ~

s_: Y= KX,

X'=-.3X+K"X} A'J -:3+~ = c.s't-~)/1<,

>
/:*O

K =-l.J'3'4d-.'U>3'4

L,~x~~'3X+KY

=·"x,

X=ce+·''tlµ; }<:','f'~ ~ ~.~~

103 ,

Section 7.4

K.=-.7

104

Section 7 .4

':1

t.•

~

7. (a,)

M>

:t'= ~

=O~ +I~

~·= -~x +~x.(1-;;)~~ ::; (~-~)x.+o~

~=(o.t~o+4~-~) )It. :::t./P-!&1~· ./21~·, -60~

~

).o

~ ~ (dl-~) ~ Mrl. ~

.rtJc. ~ ~

105

Section 7.4

106

Section 7 .4

I 07

Section 7.5

Section 7.5

108

Section 7 .5 [t,tt,~] >t=o.. ,o,

[ [o,. 05,. 05]}) ~=.03, ~=[~,~])j

~~)-{di

~ =[t,x,].

10

JO

.

109

Section 7.5

110

Section 7.5

111

Section 7. 6

=

I. (b) x;~, = - O{'x,n, + Fo CD.2:t. L:t ~ AC{.)J2.t~ ~ x;'=E-oiA+F0 )cpJ2t, x~= (-olArf;,)~Qt + B) X1:: ~fa c.oStt + ~:t+ ~ :u 0

0

112

Section 7.6

113

Section 7.6

BStJ

l.S

1.5'

2.

r

114

Section 7.6

115

Section 8.2

CHAPTER 8

116

Section 8.3

I I7

Section 8.3 I. (b)

[2.

,2-+ ,2.+(-%_),,t

1

3 -2.

o) o

•

(2.

1

o)

~~..u..t~ x2.=0; x.1 =0

o-!.2. o

•

2x 1 -~2.-x.3 -s~4 =c, ~~~~-~~- ~µ.e ~; X4 =~'> ;i(.3 =cXi.> 'X-z..=ot3 , ~,= '+5~ 1 +oc'l-+ol3 (ot>.o~)

(e)

(2.J -1-1-14 -.30 0) 11,Z~er,z+f·~)~I (2. -l -I -3 0) ~l_, ~ ei1 ~(I -Y2. -Yz. -~h.. 0) 2. 0 -Yi ?t2. 312. 2. ~2.~ -2. ei2. 0 1 -~ -3 -4

(~)

~~ ~.: (-Ii.) (

X't=
l 3)

(I

~2.J X2.= -4+,~.+~Ct!z.

3) ,3..,,~3-{2)e;1.(

J -2. ,2..,~2.-,1 -2. I -I ..3 r e;3~:t3·,J 0 -2. -I -2. l -3 -L} .., > 0 -4 -2. -q.

l

J

> %J=:

I -2 3)

O -2. -l -2

i-a1+f cti+t(·q.+J«,1+9qiz

= -2 + 3 Ol1+ 5'Cl2. ~2.~(-~)~2. ( I l -2. 12

3)

0 I ~ l 0 0 0 0

)o

0 0 0 0

-Jf

~,3,~ 0 l ,,4_,, 1,4t(-l)\2. 0 I - 1~ I 10 -12. -i- ';'I-+ (-3) ~1 ¥ . . 7 >if ' 141 0 I - 1~ ~ 0 O O ' 11 -2. ~ 0 t4 -zo ,.. O I -.1!2'7 oo 0 ..oo ~..u.t. ~: x2. = -1011 > x.1 1&,/1 3 2. ~

,3_, ei3+ (-~), f

0 '[ -5 0 JS

,tt

=

(k)ho~

C~) ~ ~

Xq.::

°'' "°!= 2.-cx.., "i. ~ (t-0(..)/2., ~,=Cl-t-Qt.)/2

c=•o: Y\o~j c=u: ~ ~ x 3 =1, x.z.=o, x.,=2. (n) ~ ~ X3 : 15/4; ~2.. ::: - 3/4, X- 1= 7/2. (~) ~M.t ~ ~4 =2/S> X.3 J/5, x2.. =1/5", ~, 2../5 (f') U.~~ ~ _x.3 :: -5/2.J %2.::: I J x, 1 -1/2. C\) ~M.t. ~ X.s=ri-, 'X..q.= OJ x 3 :-a., X.2.. = O, ~\ = -oc (M)

=

=

=

2. Th~ Ui-4 GaMAO ~- ~ ~.i.) MJ .U'.o ~ ~ 't4 ~(b)

f2

r o) ei1_, {t),r

1 1/2.

\ o -"t o i2+rt)eiz.--+ o

(IC -~l -~_,

(f)

(

1

~

~ ~~~~~)-to(~ t~ ~ -~ ~ ~ -3 2 0 2 2. -1 O 1 0 l

1 l

o

ei1~ ~1 +C-t)ei2.

~

0) ~,_, ~} +(t)~2. (J0 0I -5_, -_,~

-312. -C, -4

ci) L~ ~ o.il ~ ~ f1i. ~

(

o)

0 Y2. J -?/2. I 0 0 I -I -r -l 0

(t o o)

-2. ) -4

o1o

~ ~~ =o>

~.=o.

~ 'Xq.: o(1 ,

X3 =al?.. J

X2.= -4t&,
X-1= -2.+t~,+SQ!-z.., ~ ~-

nu.:

-+

~ ;~ ~ -·~ ~ ~

O 0 2. -3 2. 0 0 O I 0 I 0

2.

l

%.

0

l

2.

0

-l -¥2. - l 0 0 0 2. -3 2. 0 0 0 0 3/2. 0 0

-+ 0

Section 8.3 I ~O ~a.IO 1 V2. 0 o I 0 l~OOl 0 I -2. -I -2. 0 0 -+ -+ o r -1 o -2. o _.,.. 01000 0 0 I-~ I 0 O0 I 0 J 0 0 1 0 I 0 0 0 0

I

0 0

000100

0

p

0 I 0

100010 010000 oo1 o r o 0 0 0 t 0 0

118

Section 8.3 Jp,c:3,'f4.,,_ (e)

p~

~ ~ (~ ~ ~), ~.. ~Q ~ -J= ~, ?(,=-~

ru:i.

~

Clo), A.r<.

-fdt

/\=2

Mrl..

'fkt ~ ~ ~ ,,._,, -fdt fk ~

µ

(g 6 : g), ~ ~ ~ ~=o,~==o, x=ol OOlO ~ (3-~ ~ g), ~ ~ ~ ~=~,"d=f, x =2~

~ == 1,2. ~}\=I AN(_~ o..iv..d.

119

0000

8.(b) P~~'tk ~& ~ 10 x,+~:z.-4~ 3 =2x, 1 -x2.+x3 io ~;lo

~ ~: (' I

-4

2 -1

I

O)i2~:i2.+(-l)'J 0

•

(11 -2.I -4S 0) 0

~l~(O)'t(O 0 0 ____,,

l -2

0)

5° 0

~,+k .bt~i.o~.°"' ~-~~;~~~I~

o ~ ~ ro ~d'iW: ~~ ~·

11. ra.) v~: Ta ~e, + Tz ,(:wv\e2. = F H~a-r\tJ. : T, c.., e. - rr2. (¥.)e2 o . Fo't ~,=ez.=1T/2. flt~~ T.+T2.=F

=

o=o

~~~a.~~~.

-

-o

Fo'L -&l=e2. =o tk ~ ~

o=F ~ ~ M> 11:2 ~. rr1-rr2. =o Cb) V~: '111 ~45° + '112.~'10° + T3 ~30·= F en o.'1l T1 -+0.8'7'112. + o.5'T'3 ::; F H~~&: rr, ep45• - rp2. C(:){,0° - T3 U':)3o· =0 ) 0.11 '11, - o.s Tz -0.8'7'113 0. (G)

p~~ (ll.4) ~ 0.11

!f-

tk ~ ~~ ~ cx-·~p + o. a1 (- ~r.)< ~+ "3 ~ ) + o. s(- ~3 )
=

om ~(~-~) - o.s(-fiX«-+.f3"1:1)-o.1n(-t3Xtff x.+"j) =o. Ve.~~~1"1~ ~~~ ~~i4~..:Jo(11.4) ;to ~ 'r,/P2 /f3 .

JS.

2.

1'

p

'd

'X."'4l·Dl4~ ::Q d

~l~ _1.014

to

1:01

~-tk~t.mQ (~ ~ tkt ~ ~,~)

~ ~ fo.n V\\fN._ ~ ;x ~pk,~,~i-4

~ Li ~a L2.

AA<.

L2. ~ ~+ \.014~=0

~~ f~·

Section 9 .2

120

CHAPTER9 Section 9.2

1.

(0..)

Cb)

(,_A

(C)

~-.8~ ~

-~~)

~-f +3~

-

36

(cl)

--

B-A

(b)

- - - - / C z l.LI 3A I I +B I

I

~~--------- ,c~-A+B \

(V

\

-

"-I

\

\

\

'

\

'B

-

-A Cz

(c)

-

/

-A+!.B

_,..,. - " - 1 -

/

/ /

//

/,

,

7

/

B

/

/

L - - - - - - __ ,____ _ _ _

A

3.

-

CJ.-)

I"

~v:E\\ ' ' \.

/ '

I

'

I ' '

v

I I

l~

c,..., ~ J.47A-1.41B - -

Section 9.2

B+§=Q ik .t\+~+c-m=Q+(-m, B+!?=~+<-§), ~=-.§, M ~ A~s llJ1t.1 :'"\....j ~ II.Mil "tf.~ ~ dl.~ Mt. ~ ~. 'T'-lu ~ ~ ~ ~J 1j ~~' AO ~=~=Q.

7.(r'-)

~

122

J

(b)

c

AB+Bc=Ac~~foi.AB }

J)

BE=~BC ~~-

ABt BE= A£

fAE-Ac=fAB

A

B

~AB

BE=°'Bc.

~ AC-8C+BE=AE

sAf -AC= J. (AC-BC)

r

2.

~~~BE) E~a!t_AE AC-BC+~BC =AE 1 ~3 I pAE+fBc=!Ac J ~ AC-(1-0!.)BC =2~AC-2f8C

(1-]13 )Ac = (1-a-$)BC AC,BC4n~. ~ ...1 ~ 1-~ =O } -u·~ ~

8.

~-

z~

1-«-...L =O 2@>

~ex=~

3

~~= ~8+ 8~ = <"A+otAC ,..,._ ::: fJA + ol cec- (:)A) =(1-d..)OA ,,,.,..... + cl OC · -~

#1".J...,

,.,,_,.,,

,.,,.,.,,

,..,.,~

10.

:·~IJ~ : ' ~~)~ n12. ~ ~ r4-'4 ~rr1, ~ 'd- . . ~

X

..

W/.,A.O:

'

v A" I

'

I .

'°° A~A'~ F\

1 '

~

A"~ ~ Cl,0,-1) .

1T/2.

~ 'd OJ'/io ~ 1T/2, ~

t.

'X. ~:

Section 9.3 Section 9.3

t. & ~

=

(5)( s/~30-) cp('o·+w·)

( b) ~ • ~ :: ( 3)( &,) c.r.>O 18 (C) ~ • ';! :: (~X") Cr.:>\~ -3b cli) ~ •!:! (4/eo3o· )(4/~,o·) ~'Jo·

= -2s

0

2.. (0..) !Y"·~

:::

=

=32.

= fllN""llll~l\ea:>-6' ~ ~,g-:t.~

,

::: ~·~ ~

l 0 , 1 No't..U.:d (b) ~-~ =f ll~lli..epo~.. Y-*~ ; o-4. ~y.:#=Q l 0 .u.::'5 : o ~=O ......

-

(c.) (c:it~-t-(3!1)·~ ::: ol fJt.·~) + f'C~·>!:!) FcJt ol=(l~I: (~+~)·~ = ~-~

Fdi ~ =o:

~)

(!)

(ol~) •':£

-

~


+ t;i"· ~

@

=ot(~·~)

~ ® A-J.@. ~) ©

(ri~... ~~)· ~

1d (2.).

® a,,,,.J..

.

® ~ © J,..c ... 4.t.

= (cl~)·~+(~~)·'::[ fW1- ®

r~.~ ot. M~·;p:n~+~~®~®· (~+~,·~ :: II ~+¥1\ ll>trl\ C!{:ll(6-'t)

F>.Nt @:

= ll~+N""llll~l\( cpecr.>?J+~e~ t)

.

= ~u nw-n(~ n~~co~ +~ l\!fllAW.~) ,._, Jl - \ l~ll ~

N~> ® =

=ll~ll [11~1\~+ll~llep(~-~)] )./ ~ ':!;.·~+t;f·~ =u~\ll\~ll ~e- \\~llll~H Cef.)[e-~) ~ ~ ~o, (oi.!:f.)·~ =lW~ll J\~ll Cf:>& == o<. l\~lll\~I\ ec>e ) v .W-) :: H.t.t ull MrJ\ er.:> & ""' ~ d.< o, (o1.~)· A/! = n°'~ 11 u~ nct:> cn--e) s

0(, ( M •

o(.

= ICX:l ol(~·~)

\\~llll~ll (-c.oe) = ol Jt~ll J\~11

= ol u~ nu~t\ epe

co.:>6') v

~Mrl. AAt ~-

3. (~+~)·(~-4-~)::: ~·(~+Z,) +~·l't[+~) ~

® Jrm ..

={lef+~)·~+(~+'.1<:)·~ ~~~ (&~2.a.) = ~·~+~·~+w·~-t-~·!f f-"l®Jm.t ·. = ~·J!f +~·~+ rf•J.!!+ ~·~ -pui. ~ 4. (b) """"" BA· ~p = BA· (~A+ AD+ t AB+ t_At:J) ~ :DC.:: AB ...__ !. :: o+o-tc1) +o =-Y2(c) AC·DP::: 1-f'DC-t~A)+~·(OA+A
123

Section 9 .3

124

<-r) BC· (5-P::: BC· (Of\+ AD +:DP)= 0+1+o=1 (~) ~·CJP=

At}·(teA+tAB+A'D)=-t-t-O+o=-1: cft) CP·DP =('[Ae+tBA)· (OC+f P£5'+ tBA) :: -tfJtJ· (n:+4:ACJ+tBA)-'"t EA· CDC ... -£AC1 + t BA)

c0'XJ·

= O+t(O)+t(-l)+O+~=o,

a.A'eovlJ,~ ~ ~ ~ ~

p

~°'~c~o:r~~)1tko='~· 'D
5. (Cl.)

~APO= ~· u:i1~1i;gu

=~· (~A+tBA+!DA)·(tAO+fBA+'.DA) n

= C{.)1 -i +ii: +I

·fif+ f +I U"+ tf-1-1

(h)

~

APB -

- 1 PA· ?B

qp

_

{C)

~ Afc

=crS""' JlPAU PA· PC = ~I 0 JI ?CU

(cl)

~Af;D

=~' UfflllllP.DU ffi•PX>

Ce) ~ ABf- crS1

-

C1-)

~

Ap C

=

_

-{f{f -

·)

48 I

-1 2- _ o ep 3 - 4 u::i

_ -1 -!--1 _i_ _ o - c.o ~.ff = ep i3 - 5''1-."74

BA·BF

i-

-

-1

=

-1 J_ cp tfi

llBAUllBPU - C<-> (l)l{.f

.. , cF\· er

Oto llCflllllCP(I

=Ct->-1 ~ = ~5'. 9 I t

ff =

-1J_ 2

cp

= (,O

o

o

-1

o

..L

~

~ Pl

V

=

_, PB·°"' r ..Y l\ PB~ l\ P.DU

C,,o7:)

:

-a 0

f:.tj

-1 I

o

:S::

90

o

. '?!

= 90 °

cc.i

(A..

..

ffi· PO -I J.. --a = ~ llffillHreU = ep .rh=f =4:> t =rio.S3 '11:""1~ ~~PC = ~I PS·PC ~ 4° lll'SllUPCll = ~;rf' =<:r.> ;ff =5 "''7

(~' «BPO ({,.)

- cr_;"t 2. 3 -

-1 _L

llPAll llPBll - cp

nn

..

IL

u

Section 9.4

Section 9.4

125

Section 9 .4 Cb)

g~, =: ~

126

4CJ ,.._, + Ci,010,0) ~ + Cl,0,0,0) ~ l&O.t) n,o,o,o) ~ CJOtt) ~ (toe)

=

=

Ct)(3~):::. t c1,o,o,o) ~j i,~ "'d '°b~ 1.~

~

=

t°1- (1oe)

..

=

=ct)

..

~ (10~) o,o,o) tv1 c~h)

{C) ..U- 4X, ::O

(~-4~)+4~ :::~+4~ ~ ,,~;IQ~ ( ~+(~))+4~ = 4:Z +~ t"1 (~e) a't\LHS AMA. t{o~) 6n RHS

tm (tob) ~ c1ott),..,, LHS ~ noc) ~ R~S

~+ C4,e,+C4~)] =q~

tm

~+Q:::4~

~

C1od.)

=4~ 1AA- Ctoc.)

.

l\:"~=¥4~) 4~ ~~ ~ ,,~ C¢4)~ ~ ct"oeJ

=

::: .i 'X.

= ~"-tu' (lO~))

4.

(Cl) ol, (.2.,L3)

~ ~

1

1

-

tJ.Mil. ~~~ ~i4 ~.uf.~

=

ol, o1, = o/3 = o.

tt

Wt.

{~.b)

°" +2.ciz. + ol3 , 3a -4cl2. +ct3 ):: (O,O,O)

2.C\',+ olt. :: 0 ol1+2.0i.z.+ol3=0 ~,-llolz. + Q'J =0 2ct I

={0,0,0)

+°"2.(l,l.,-4)+ oe'3 (0,l>l)

( 2ol 1 t ol2. J

C.b) tl.o. ~ c~),

=ct,* ,o,-t )_ F

2.0(13 : 0

ol1+ct2.+~3::0 ~~~-r~ i.-~81-. . sot, +olt -~3 =0 ~ of,=c, o
(C)

Q.o. ~ (D...),

~ ~~ 1 +ei'z.-2.cV3 :: I o{, + 2~2. + =3 3o!.- 4cli - ~.3 =2. ANt

"""'°- fu..wx> ~ ~ ~.M.t ~-

°"3

ol1 = 39/2 6.

(cl) ~ ~ {~) J W( ~ 2ol,

-2.0l3

ol, +ol2.

3ol, (e)

+°'~

D..o. AM C~),

+ ol.3

-

o/3 = 2!}/2.'i, olt = ~/7,

=~

=0

~il ~ ~. ~ ~ ~ ~ ~ ~~ ol~.

ol.3 = -J ,wt ~

«,

~.+ol"L

-4ola

C1) ao. ~ l6.), t.vt ~ o1.

=o =

+ oli.

O

=0

~

4..u.t. ~ o1,=

+ 2.a3 = -2.. ~ ~ ~.

2.ol1+olz. - ot.3

=o

-lf Ol, -r O(.,_ + ol3 = 0

Ol.2.. =O

·

~A.At~ d.

3 =-M1,

o/.2..:: -2./'7 J ol.1 =-2/7 .

Section 9 .5 S.

No. E.~.) { ·~=c,,o,o,o) a..v.4 .~== (o)r,o,?) i{k/\. Ol,.~+
127

(a.)

1

o(, = CXz.

Section 9.5

=0.

i1

Section 9 .5 (j)

<;: GHJ =

Gv.J

128

~\,~~;,~[;J"l\ =~· f-31-~M,-4,-~2 = cP~:!{SO:: :s~.1·

H ~HJ&:: c,:;'\~;l[~GU = ~· cs;1-~~·1:t4) =cO'~~ :: 45'.~· ~
S.tvm.

4. Cd.) ~=-if3

s.

= cn- 1 GJ·GH

)IG:J 1111 Gtl l\

=~I (-2,0,-4)•(3A,-l) ::; ~l -2.

=3~.1 + L4S.~+ ~s.o = 100·

(2,Z,2.):: (V.[3) 1/{3,

~/"\ £

~

~s20

t/W '6Z

= ~s.o·

v

1/./3)) &'=*7-(~4,-5,-'):: (-it/ffi,-s/m,-,/ffl)

s·£. =- <8-§)· £B-~)

-

B

ci. =A·A-A·B-B·A+B·B =A2 +B'2.-2Al3Mc::t

~. (d.) ~= (1)2,0,1), ~=(1,0,1,t), ~= (2.)-1,1, l ).

u ='

,~,e,"" - ' -1.A - - N )UC.~1~-

Jt. ~-

-:.-

~· ,._ rv:: 3 >.w-· >- i.r -

--

2..' M.·w-= I > N'"· )JS'"= 4"'"" G,+2CX.=OJAO ol=-3·

M.• f\T=

~ 2 ·zt,=(~+o(~)·~

~3·~,== (~+~~-t~~)·~

='+2.(3 +)(=o

~3·~i.= (~-t e~+o':f)• (~-3!f)

: (, _') + ~ (2.-') + l{ ( 1- I 2.) ©~®~

o..k.o) ex= -3,

=7
=-~ p- ll if =0

@

p=-l'>'f-/3S-) "t=l4/5. ~.

=~ =<1,2,0) a)>

~'Z.

= ~-3~:: (-2,2.)-3 ,-2.)

¥3 =~-~sq.~+-~~= ct,-~,-•,-t)

(d)

~:

(l >2.) /

~: ( o,?..) , Jff: (Ir I )

VJ"'"'Jl~ ~·~=S") ~·~=4,~·~=2. ~ •~ ='f) ~. ~ =-I , t;£. '!! =-2. ¥i·~1:: (~+d~)· ~

=

S-+ 4c{

=0

) ot:-S/4

~3·~·= c~-+~~-t"¥~)·~= 5+4(.3a-~ =o ~·~t- (~t~~+a'~)·(~-{ff)

©

= =(S--5)-t-~(4-S-)+
= -f+~?J=O ·©~

@

~ (3=-3/2., ~=-I. a.k>, c:t=-5"/ll-) ~. ~ (1,2-) ,u~ u.-E.f\r : (I , - ~2..) "- 4M3. ::: ).J.,-~~-Mr: (O,O) M~ '<.tl\.O~)

®

= =

=

"'

""

2.. -

-

o- -

Section 9.5

129

+;.!l

CAlt"-~ 4~ D.M.. o"CfJr~.1. ,~. 7. (A.) ~-~= (-l,"3,-4,-t) , ll~-~ll = ~t+~+l" +I :: tf2?i =:3.(3 AO )Art_

(c)

8. (~)

*u ~

H

=.1. fort. ~ ~=t:Q ·

11

--

flM+tV}fz.t lll.\-fV"ll?. ,.,.,,-

-- ---1- -- -- -- --

=(.U.+l'f)• (u.+l\f) + (.u.-~)·(.U-1\r) --

+ .U • M. - 2 U• l\J + t\r• l\r = 2 ll.U. !l t. + 211 f\J l\2. ... ~ ~. (J,) ~ • (2,1, I)= 2.U, +.U2. + .l.(3 :: ~ ~ ~· ~ .U3: 30) = Lt,-2u., =o_ ~_..J,.,;.,... ~ A.t 3 = '7~, 1..-lz. =-~, ~· -2.ol. ~· (2.,3,1) = 2.U 1+3M2.+M.3 -0 MJ ~ - o<(-2.J-I> '7) fdz. ~ e:{; ~.e., &.. ~::: M.•.tt + 2.U. • 1\1" + t\T• f\f ~-

1

~· (1,0,2)

+2M3 =o

= i.u.

(~) ~·(2,V·\)= 2.U,+.U.z.-.U 3 =0

=

~· (J) 1,1) = M.,+ .Uz. + A..l3 0 ~ • (3,'l,t):: 34, +~ + u.3 =O

10.

(~)

.U. ~

=.lt +.ll.z. = w

1

-

o(tV'" + -

Mz. ~

I)[·~: olll~llz.+~ 0 ,. .u.

.AO Ma

-

=

1

~ ~.

.. . ... ~ fk ~ ~. o/,=ol.z.=ol.3-:.0

r

~ ~=Q.

r

A

ol= ~·~/ll~lli. =~·~/"~II ,,. " olrv :: ':[• - ~ ::: (/\f·M.) At ~ ..Uz.= .u.-u., .

=

l\i\ri1,..,,

~

rp~ ~

=

-N~

LJ.cl ~ ~ fort

(f) M (2,1, 0,0,3) 1 ;o... f\f": (.l'D I oI I1 -2. ) I) ,._A

~= Jt (0,0,1,-2.,1), &-·~ = -a/,Z

AO

""'-...-

2-

~ 3-~,

~· =<~·~)&= ~

OJNl ~2. =};d.-~·

11.(a.) («~)·~ o{(~·~)

Mfcn. n-~

t

(n~2).

(0,0,1,-2,1) = (o,o,t ,-1,-t)

= (2,1,-t, l,t).

=(«.t.t r .. ,A-tn)·(rv;, ... ,l'Vn) =OlU.,l'J,+···+ct.U.r/vn )ikt~v 1

= ol(.U 11\r,+···+ .LtnN';,)

Cb) (~+~)·~

= olU,Afi+···+

=(.U. +NjJ ... , .Un+Af"n)• (MJi>... >~ J

ciUn~

1

=

(.U 1+1\fa')A>J\+···+(Un+N;,.)WV\.

(C.) (~+ ~~f)·~

= AA1 MJ&+···+Ltn~

-

+ fViMJj+···+ ~~ = AA•Mr+N"'•W --... "' ,...,

=ct(~y ~) :r (3(~f-¥)

ol=p::.1 ~~~Wt,~ (~+£r)·~:::g·~+![·~

r=o ~ iL.. ~ ~ N~~ii.t~:

{o(~)e.!t1' = ol(~·Jtr)

(ol~+P>~)·~= {ol~)·~+(~t;t)·~ f"\ ~ :::

M°'4

o{(~·~)+(?,(~·¥) ~ ~ ~

-

v

12. Ctt) ! 1== epct. =u,/n~l\ =Ll, /4°.uf-+Ml+.u~ .Qz. =cr.:>fb .u2. /n~ll = ,u:z. I if " ..Q.3 ::: ep(f = U3/l\~ \l = J.J.3/ ,[ " c&.> ~ = l't,0,-3) ~ .e. 1=4/S", ~z. =o, 13 =-3/s (e) .Q, +R~ +Q~ =(.u,/,f u~+M~+u~ )'+(u2./f.u~+M~+..u~)i.+ ( M3 /.[Ll~~JJ.;+.uf)-z.

=

'!l

::: (.ur-f:'.ll~+l.t~'/(.u~+Mi+.u:)

=l

Section 9.6 \3. No. Fen..~.J

14. rs.

Cb) (b)

No

'1

130:

~::~=0,0) ~&.~-==(O,l)>~~·~:::.O,t;(•r.f::O, ~~·~=l~O.

~> Y~

c~) 'I~

+x2. ==o) ~ ~. ~ x3 =ol,x.z.=ct,:>G1=-oL ~ ~=oc.(-1,1,1) x 1 -xL+2x.3 =O j ll ~ ~ ~ -ti-.:t 1-:.t 1-~ ~ 3 (-1,1.1). cc) x,-~a..-s-~3 ~ ~. ~ x 3 =ct, x2.:=-~+~, x,- G+«. ~2.+4X3=6 AO ~=("+~,,-4c<.,ol).~ o<.X, L ~i/\l,.4.~~~~,2t-~ ~o ?t

1

±J

=01

~~-#...~1~JitM>~-t4

~~~~{~,~~~. ~ ~ ~ J-t(A, ~ : ("+ ~) C,-4ol,ol) (,,,Jo)+Cl (l,-4-, I) a ~o + cX ~.

"-S~

1

= • rr~~ ~~ L ~ c
l.

\l_-~1

,..., oC :2..

z

~,Nor(.~ ~ 4- ~ J.,,.~ L ~ ~~ :iO ~ ol'r;,,. ~o'l f/X.) 1

d:: (8,-2,2)-(7,2)1)=<1,~4,l).~,~:io ~o.A_~i"l.. ~ + J...!.ttu_ ·- ~ -. () ···--d-.. . d . (1,-q., 1)) ~ ,,... -~- -- .

;..., ol=I

-is

NOTE: I believe this simple problem is a good one to assign and discuss. The common error is to simply take the resulting x vector and scale it to unit length, whereas the x vector is not along the line L (unless the two equations happen to be homogeneous). The key to understanding is a simple sketch, like the one above, and it is important for the student to understand that it does not even need to be to scale, but only schematic. Generally, students do not use sketching and visual aids enough, and this example offers strong encouragement for them to do so.

Section 9.6

•

Section 9.6

131

(c)Y~

Cd.) No,~~~~~ it~~~ ~~a.et C~J-~~~!): (l)

fa.k, ~ ~+!f =(1..ti-..rl) ' ' ' I

.Jl\-N"',,)

~

>n.t~ ~

J.;,.. ~

~-~= (rJ;-u,' ... > l'JV\-J..(.Y\) ~ ~ (~+~)+~ = (JJ. 1-TV; , ... >Un-~)+ (A.>Jl, •••

(2.)

=

~

(.U,-1'1'"-Mlj

>

> MrYl)

... ' .Un-N"r\-.W-n)

~+{~+~)=~+(~-w{, ... ,~-~)

Mtk~,~~a.t

::. ( .ul-N;"+~) ... ) ilY\ -rs-Y\ + w-V\.)

t

a.k,~~·~ ~~~(4-))~~J ~ ~+(-~)= (.U 1-(-A.l,)>···, Ltn-l-.Un))

.

(O{+~)Ltn.) . ~ cl~+~~: (dll 1 -~, >... , clUn-~"'-). (3) ~ ~ u+O = LJ ~

a.Joo)(')µ (e)

No:

~ (oi+~)~ = ( (ct+~)J..l 1 ,

= (2U.,··· ,2Lln)

aioo, ~ ~

... ,

1..u. ·

;;t~a. ~ ~ -~ ~~ 1Q .i.o..c.o.. ~ ~

tk.t ~+(-~)=Q) ~ ~+~=~ fd1. ~ ~ r;f

~' (&,)

-

µ

~ ol~+f>~= ~.

.

2. YwfAJ, ~ ~

~ (~+~)~= ((ol+(!>)U,, ···, (ot+ts)Ltn)

(2)

~ (~+~)+~: (..u 1+21\f"1J."> .UW\+2~ )+ (A>Ji, .•• , >..Un)

= ( .u, +2~ +2...vJj )

=

... ) .u)\+ ~ + 2.W~ )

~ ~+(~-+ ~) ~+(~+2~,. .., ~+2~) = (.u,+2~+~~, ... , .u.W\+2~-+4W'"r\) a.!Ao, -~=c-u,, ... ,-.u~) ~ ~~ ~+C-~)=~ ~ ~+(-~): ( U 11-2(-U 1) ,

••• ,

.U.n+2(-.U\'\))

1 , ...

,-Mn.):-~

=((ol+~)u 1 , (ol+~)Un) o<.~ + ~~ = (olU +2.fY-ta) ... ) oC.Ll.n +2~.UY\.)

~ J (') ~ ~ (ot+f3)~

~

= (-.u ... ,

1

7

'It.};,~.

Section 9.6 .

i]

.

132

.,.

tMAt~ fo't ~-- : Wj fdt t'r~ ~~tJA, fVNl -"-'1 -fol~ ·'ef ~~~ . Pf. ~ ~ ~ ~ ~ ~ ~·~ = ':!·~ (t,o.) ~·!f- > 0 fit oJl ~t~ :: 0 fo"l. M :0

w

Ct

0..

J)

-

--

(o{~ +~~)·~:: Ol(J~:t)/J°)-t

1\~ ~ Jl~ll = ~~·y., ~ IJ cl~ 11 ~(al~)· (o<~) = ~ ol [~· (oi~)] ~ (IC.c)

=

f-' (r;[·Jff)

(l(.C.)

r

=4o( Jtotg)·'*J

(lkll) fWl (l4'C..).

:: 1o(l (~·~)

=JctHM·~ = lot\ l\Mll v N~t, Jl~ u=4y.·J;J > o ·~ ~:f~ "'rn (1'1->' ../ ::: 0 .M. =CJ " ,, F~, 11~+~11~ = (~+~·(~+~) =~-~+ ~·!!+!!·~+!:f·i::!

= 11~1\t+ ~~·tr+ ll~\\'° ~ (IC,tt)

~ llUl{Z.+

2-fU•/\r} + 11~1\L ·

~ l\~ll>z1i~1ii1~11+-~l~llv. t':S~ ~

~

a

ll~+~U

.

=(l\~11+ ll!ll)t.

~ 11~11+ 11~11,

(13)

AM

S.-.C · ~$

~ .v.rt'/\.l ~. v'

.

·

.

U.t~~~~~iEx~7~i4.~~Mi M ~ ~ ~. N""'", 11~11 = IA.t.1+···+1.u,..1 ~ lfol~ll = Jol.U,\+···+lol..Un\ = lolll.U,l+···+lttll.Un\ lOltJl~ll V N~ l\Ml\ >o (~:/=~) ~ ~ ~1;,i ~ ~1u·1 ~o. 0 ( ~ =Q) - --- - J ~ ~-.. a-

'f

=

) .. =

=

~ > ll~+![ll 11 (.U1+Nj, ···) wUn+~ )}l :: IM,+l'Ji\ +··· + l.uY\+~ l . Naw-, ~ ~ ll,b it'~ ~ ~ iU Jll+l:>l~ ltJ..I +l!?J) ~ II~+~}( ~ l.Utl + 1N& l +··· + l.Unf + l~ l l.Ud+ ··· + l.u.1\ \ + 11\fi I+··· +I /\Jn\ \

=

<}. (a.) J,o, 11 ~ 11 ::

= ngn+ u~u. v ~ 1JJ.~ l ~""1 ~

1~a~n

s~tk ~~ 1.u.~1 llMllik ~1ct.u;1 at:<.Wt a:tik ~~~,Ao llol.U..l\ = mo/X- l~u.; l = lOll ~ 1.U.41 = led Jl.U. U. / -

t~a~n

" >0

'~~~n

f'I\ ~ M*() NJ.4X' 11.Y.ll ='~~1n 1u; I ::: o f ~= Q

·

,

-

~ ~uA-

Nv -

--ll :-

---r-- ...;

F.W..~, 11~+(!11 = ~ (u~+41 ~ ;¢n. (tu;t~tr.i;·t) ~ la.+lol!ii 10.1+11o1 ,

~

~

~ t~n. . a- lU~ l

c-----.

+ 1~ . . ~ . . n IN":;\,

= ll~ll + lltf 11

Section 9.6

133

-~d~t~ ~tJ.;¥~·~. N~t!'4t~ ~~ ~ ®~© ~ M~'L ..:t i4_ ~ ~ ~ ) t<-6t Afl(, ""- dL ~ ~ 1lA- ~ -rkt. ~ "-A.

1

~NV\-@.

=l~r\ IU~I

~ ~ ~ (l'lh) w (l7C) OJU.. M. F01. L;t ~ ~ :: (2., I, 5') o.,,.,.iJ. t;f: ( O, ?. 2.). 'T'"""" 11~1\ =I > 11~11 =0, Jl~+~ll l\(2,4,'7)ll 2 IX) JJ~U=o .~th.tv.~ t! :f:Q, ~ ~ ~ fl7h), ~a

~ JJ~ll ~,

(b)

.=

~

z

(174)

1

=

2.~0+l ~~ (l?C)~~>~~.l')(~J~~.

JI. ~ :;: J~ >.A(?C) 11rl'll) Mr(Xl ~ ( 4A'C'X)> 0 "' O~ ~ ! I) , ~ =Id ~l'X)AJ..{X]W('X.)~ = .U('X)Ar(?(,)W('X.)dl)l = <~,fV) v ( JJ, .U.) = J~ U..?..(~) Jvr (tt,) d.4., > 0 fdl. ACX)* Q ../ =0 .fo'1 ..U.~) =0 <(O!U+@-ff»»r) = .f~ ~Ml')l)+~N'"l~))).JJ"tx)°WWrix> h.-+ ~ S~ M"t~)w-t,e.)Wr'X)~

s;

+ ~ (tr, W) 12. (a,) (~·!!I= 13+2.-S} =0, Jl~ll =4'3+t+l ={ii', Jl~Jl= 41+q~25 +1' =~) O~,Ji\~ (b) I ~·~n;:: fUo) +S(B) + 3(4) + 2(-3)( = "'") ll~ll =~I (l) + 5(4)-f- 3(1b)+2.(g) = "'87 J :

=~ 1(0)+5(1') +3(

=

v

~ 95 J 4b~ ~87~ 85.~9 l~·~l: IJ(2.)+2(2.)+3{2.)+lJ(2)+5(2))= 30, ll~fJ=~1(1)+2.(1)+3(1)f4(1)+5"(1)

I) t;fll {C)

ol
l ) f 2.( I ) :

=

=

u~n ~ 1(4)+2(q)+3(4)+4t4)+ 5(4) Wo > 30 ~ (IS ~'o (cl) u~u = ~ s~ (2+x)'L tM< =~ 1,/.3) U!!JI =~ s~ (3~t.)t. d4.. = ~,;5 KMl.J/l 1 s~ c2+x).3?C,,~# 1 = 1114 ~ tfiWf,15

=

IS. 1"]1-lct ~ ~

1-f-4 ~

~.?5 ~ 3.3?'

=415,

V'

,

v

an ~ 1 + ··· + 4'1~ x.n =o :

=so

v

!

©

t{rr.,X1 + ... + amn X.n =0

1pk .c.J:,.2~, 1J1~~ ~ ~ ~ ~ ~1~0.N-l ~

o.nt

1

~ (!)~~AO~+![~ Dn
..

.:

lJcn1(.U,+~,')+···-+ a.Mn{ Un+~)= (a.l'Y'al~1+··· + a,M.Un)+ (a,"'

..

flfi+··· t

aD\n~)=O+O=O ..!

Qk 1 ·.t~.i'4 ~ ~ ¥=Co, ... ,o). ~ ~·~~~~ ~ ~ ~ AMJj ~.+~=CM,, ... ,)..(,,)~(!)~-~:~=. ,

(-~ 1 > ••• ,-u.,,,) ~~~· ~~Ml~ MM.~~ ~~-f ~ cu,, ... ,.u't\) ~ © ~ ~ ~ 0\'~=(~4,,. .. )cl.t.t") ~ l\n {olU 1) + ·• • + Qan (cl.Ur.) :::: ol (all U, + ··· -t- ain Lln) = ol lO) =0

=

.

Section 9. 7

134

Section 9.7

r.

~=ct,~,+
(b)

tJi

=

Ol*

Ni=

Ol3 4- ol.IJ.

= ol2.-\- ol.3 + ol4 N""' = «, + ct~+ ol3 + Olq.

~3

.

.

.

'5'~, ~ ~ 1>~ ~4.

(c.) ~:.~,~a+···+ °'S'~S'

r ""=

+ 2a'z. "'2. = 2a, + 3olz. + o13 /\13 = ct,

/\Ji

2. 3 I 0 I nfz

........,.

Ao~ M:t

+ Ols-+

~5

+ 20/. 5" + 30{4.

0-1 l 0 -t /\f2.-2.f'Vi O 0 t 0 I f'J3+N;.-2.tJi 0-2 g N4-'rv~+i4rJi

o

0 -~ I 0 ... , tv4 -tfl'Jj

[R;'

f'Jl

t2.00 I

OI002.N"'3

OI002./\J3 4 -I 1 0 3 t-rq.

.

=

12.00IC'Ji 0 ... l I 0 -I tr2.-2f'Jj

_..,,

da

rV;a. 4ol,- ol2- + ~

~~.~ 12.001

~ fdl. ~ ~ ~

Q1\t.

4

o

12.00lrv. 0 -t l 0 -1 l'f2.-2rJl

0 0 l 0 l

l'J3-+ r-.rz.-2Ni

0 0 0 0 '" N"q..t' gAfj-1\)2.- 2.Nj (e)

~ =d, ~· + olz. ~z. +

ol3 ~.3

r

"'2. =

oCt, + 2d3

~~. ~ [~ ~-s~ ~1~r~~I;:~l~ [: ~ ~ ~~ -3-' oo o s 5

3

I

o N"3-rr, l\f3 + rv2.- rV; ~ ~ ~~-~ AS3+3~-N;=O. 'f1~ ~' ~,/;j 2 ,~.3 tlo NOT~ fR 3_; ~ '1'tNvt~ ~ ~3~ f~ ~ f'J3+3N;-N;=O (f) YJA(ft.) Y~ li.) Y.io, c~ > No l -k) No Ci) No (M) Y.14 en) No .

l -l

-ttu,

1

ceJ> Y~ 2..

(4.)

rU"3

L

Cf) Y~ Cb)

cv Y.u:v 0

(G)

)!_

3. (b) No, j: ;,o ~ ~ MN--cluL ~ ~-. Fdl~~, ~+~=~>-0~~~~:

(c)No 1 :t~J~~~~~

~i ~a_ ~~-~~-J:

Section 9. 7 4.

135

(b) ?G 1+ x2. + x, 3 - x. 4 =o. x." = ot., ~ 3 =p, ~z= i, ~.=of-~-~, /;>O ~: (o( ... ~-?I, 'lf,(3,ot.): Ol(l,O,O,l)+ ~(-1,0Jl,O)-+ 2'(-1,l,0,0)

rr.i.u:, ~ ~ ~ (~ ~ tt "~",.,;,.. IR 4 ) ~ ~ f (1,0,0,I)) (-J,0,1,0)} (-1,1,0,0))

(C)~~.:

rl

.

~~z.=0,~3=ol,~ 1 =-ol,~~=(-oL,OJol.) . ~~~Na. ~f(-1,0,l)°j,

-1I0]-+[\-110]

I

I

ce) ~ ~.: [l ..,

I 0

02.00

. J . I • I '• ~A.a. A.~

o]_..[ I -I

·

ID!

/..M, ' " •

I -2 0 OJ ~ Xs=Ol, X.4=~, x.3 =20l+3(1) I -J 0 I 2 0 0 0-1 .3 2. 0 'X,2. =?I> X. 1 2P> ... (2Dl.+3~)+ ~ = -.2ct-{:>+~ ~ ~ = (-2ei-(l+tf) Cf) 20l+3~, ~, ot) . oc(-2/),2.,C,l)+t9(-1J0,3,l,O)+ ~(1,1,0,0,o)) AO~ ~ ~«.ct. i.o ~ (-2,0,2,0, I), (-J,0, 3,1,0), (J,1,0,0,0)}

=

I -2 0

=

f

(1> s~ ~ ~ ~ f c-3,2,0,1),(2,-1,1,o>J S: Cb) ~s=a, x 2 (3, x. 1 = 3«- i (3 Mt ~= (3ol- ~~, ~' ol.,) = ct (3_,o, 1)+(3(-~,1,0)

=

(3,0,l)~cl (-t,l,O) ~~~.

M

·

NO'l'E : W.W: {=th ~~ k- ~~) ~ .2.~,+ ~i.-b~ =lf '?

T~ X3 =ol) ~2.=~>

AO~ ~

Ao~= (Z,O,O)+CX.(3,D,1)+(3(-f,l,O)

X, 1 : 24-3o{-t(3

A.o-'f4 ~:

(3,0,l)

~(-~>I,~)~ tJ-t ~· T-k

~ (2,0,0) ~ ~ ~t4 ~ f)(( )

+ ~( ) ~

)A(1,

~ ~ :10 ~

~ p~ ~+kt~)·~~

µ,

rr- ~ ~ ~.

This variation might make a nice examination question.

~. (b) ~ A
.

~

.

~ ~ (..t.e., ~) fo"t, ot,{3 -f-crt ~ ~ ?1) $ ~ ~ ~. LX Mo- ~ ct;,~~-) o(+ 2~

2.ol.-

3ol. : ~

..

= "g+ 36

P' =

2(5 + ~

=3 i! + Sb

~

~~o/,f>.~J {itAO~:k>,ka.~~~i-4 ~

ol.,~>

M' uo ~fen ~' S. ~: ~+3~ 2 ~+3Sl

l 2 [ 2. -1

3

I

l

[I

2~+~ ~ 0 -5 3~+5b 0 -5

[I

-SS

2. ~ 0 -5

-4S

O

~

o

1!+3~] ~,~~~ ~~. -s~ ~

~AM~)w ~ { ~=o. 7'~) l't;:9,~ ~

~AA(,~~.

(c)S~~~~(b).F~,M~~({~) 4~+(2>= ~+2S } ~+(3= ?f-&

f = i$-2s

~ ot,(2>. G~:

*

-

Section 9. 7

136

s..rr~ ~ ~, ~ ~ ~ ct) s~tkt .R· =~ H.1., . . ,~1t 1 ~ .i.w-.A..ti. ~. >JtL~ l!=«d:fr+···+ clR~*~ y =~·~·~···+ f>t-'*t k ~ ~ ~ ~ J'. 'T'~ . V+Y = co<,~,+···+~k~-tt)+(@>d~la-t-···+~k,~) = c~,~ts·)~,+·.. +c~~+~1t)~t) ~ ~ ~ i' ~ it AA, °'- ~ ~ 1' Mi, ... , ~t: . v OJl{_

=

.

Ml


.J' ~

~ ~}}NJ~~ 0~ 1 +···+O~*:: 0~ 1 + (0~2.+···+o.g*J = (} ~ ( .. )

11

{15A)

~ Su:..9.,

+··· + O~-fc- 1J l3) ~ S.tc.. ~.c, =···1j~t4~=~i=~. /

=

* 'l'o ~tkt ~,

O~z.

(2!:!-+3!:f")+ts~+4~) =~+[3£f+(s~+4!f')] ~ c2) J.M. S.t.e. '}., .

= 2~+[3~+(4~+5~)] ~(I)

''

= ~+[(3!J+4~n+s~J

..

: 2~+ ('7~+S~)

= ~+ (s~+~!f)

=(~... Szl)+ 1~

= '7~+ryi~ rp~~> (.2~+3~)+(S~+L\-~) ~~= (2+S)~4- (3+Lf)~.

~(2..) ~ {')

"

t;

ll)

..

~

(2.)

IC

~ {')

..

Section 9.8

137

cm) i'~~· °' -~ U _a____ ~ -U -U - .fcrt ~ U~-9'~ r

,.,,..

\

I

,.-.,,

"*

'

(cc'a-!d-1+···+a'k~ic)+ {-ct 1 1-···- dk.~'k)

= (ol,-ot,)~,-+···+ (ol1t-OC.ft. )~-k. = o.u., + ....... o.u,.._ k. :: 0-t-··· .. O ::! CJ v

-

[AA~(.\)]

civ)i'~cW ~~ ~~.-~ ~ · olTJ = «. ( ~ ~ 1 +··· + oq~~-.) = c£, [ o( I~ I + (ol2 ~ 2. 4- ••• + oc't. ~~ ) ] = ol(ct1 ~,)+ ot(otd!z.+···+ ol-t.~~) c1)~ S.tc..~.'- ~ (olol, )_g, +cl L (o{2.~2 )+ (Gt3~3+···+ ~~«)] ~(5) AM S.ac. ~.Gt = coeot,)~,+ ct(a2 yi.)+ «(ot3 ~ 3 +··· +ct-k.~-1c.J ~ ci)~S~·'·" :: Cota,)~ 1-" {olot2. -i:{t) + .. ~ £5') ~ SJ.C · '·'

-'?

=

. =... =

(olOi,)~·

~~~~ J~

+··.

../

* {Ola!*)~*-

)

Section 9.8 2.. (b) g'"j ~ (a., g)::: ~(1,4) +0(3,,-1). Cc)

..

eel.) S.J;

=-3(1,-1) + Q('t, 2.). a.(1,2,'3)+b(s,2 1 1)-t-cCs;S,~) = ~ ,dl.> ·•

a.+ Sb+ SC= 0 2.4-t-2.b+s-c=o 3a.,+ b + sc = o

(-3, 3)

[ I 3 5" 0]

d. =lf, ~, ~ s~ fen. c1,'2., 3))

J...J

.s. Cb) a.c1, 3,

[J3 5 0 ]

[I 3 S 0

1

so-+ 0-4-so-+ o-4-5 o 3 1 s- o o -B-ro o o o o o

~= 2. 2.

=

=

AC C Ci, h -S'a/4, q_::-5«-3(-5«~4)

=-.sot/Lf- 1 ,(K)

wt

~ -sc1,2,'3)-5'(3,2,l)"'"4l5',S',s)= ~. ~ -e,<~, n,2,3) = ~ (s,S", s)- cs,2, 1) . + bC2,o>-+ c.(1, 2.) + tt.c-1,s-> = S2 G\.+2b+ c -a =o 3Cl+ 2.C-+ S"cl. =0

o..:-5",b=-s, c:::4

F

cit)~~~·)('

2 l-1 0)

o-,-1 '

0

.

,t.(j

tl= cc!) c =~, b =- t(-3 + «.) a.= O{-f?'+

.tca-2a.

=-C{,-t~· L..toc=o~~=,,~.lk cl=o,c=,,b=-1>tt=-4- A-O. -4Cl,3)-C2,o>+'c',2.)+oc-1,s)::~. s~~ +rz, c2>0))~~~, ~ {2,0):: -4(1,3)4-G.l1,2)+ 0(-1,<5). L'D. Cc.) Ce)


N~~~ ~.o.c-·~ ~

~LI.

L1? ~) ~ ~,

1-+4 .:tb.

2.a..+h+3C=O }

2h-+ 2C.

=0

?'P-4.9.e.2., ~

CO,D,2):-j-(o,0,3)+0(2,-1,s)+O(l,2,Lf-)+0(7,,9.,l)+0(2,°'-q.)

w~ cA.(~,3,o,o)+l:>C•J-s,o,2.)+c.cs,1,2.,2.)=~

sa.-sl:>+c.=o 2c=o

l>l:J,

,o'l.,

~ ~ c.=b=tt=o,AO ~~ "'4.

.

LI.

l[ 4 l 4

Section 9.8

138

(~) \J~ tt(f,3,2,0) +h(4, l,-2,-2)+ c(0,2.,0, '3 )+ d.. (4)1,I,2.) = ~ , d't J U.:t· 11-h +4d =0 sa.+ b+2.c +'7Gl.=O 20..-2.l> + tl -2b +3C + 2.d.. M>

(~)

=o

~

[Io-n40 -so 40 2.

~

o -loo -'7 o

=0

l

4 0

0

o -n 1 s o o o ·irto -~II o 0 0 ~ ~ 0 ll &I

0 -2 3 2. 0

a.=b=c=4=o, ~LI

~

J

o-n oo

0 4 0 220

0 0 2.'J

I 4 0

4 0

s o . .,. o ..u 2. so 2.7 o o o 20 2.7 o 32.

0

0 0 0 l 0

rn+

~ fdt. ~ ~: ~(~): A:= ~re (a.,1,4 ], r:o,2,-4 J, [1,o, s], (-J,3,-'J, ro,1,-a.J J )i

Ld'.(l,

b:= ~ (Co,o,o,o,o]); ~, fdl. ~ ~ a..(21 0,l ,-l,O)+ b(1,2,o,3,l)+C.{4,-Zf-,3,-,,-2.) = ~ J ert, 2.ct +h+ l#C:: 0 Ott +21:>-Jic =O 1a. +oh-r3c=O

1-

=o oa.+ 1b-2c=O,

-J0..+3b-~c

tht. ~

.

,-r.

(-3_t 1 , 2_:t 1 ]». 'VJ~ ±1=1, ~' lr.rl. ~ a.=-3,h-= 2.,C=I, .o-0 . -3(2,0,1,-1,0)+2(1,2.,0,3,l)-1-1(4,-LJ-,3,-C},-2) =~) rn) ~ ~ ~ ~ ~, (4-,-4,3,-~,-2) .3(2p,1,-1,o)-2(1,2,o,3,1). Cl) Ll> ~ 0(1,3,a )-t 0(0,1,-1)-1- 5lo,o,o)= ~- C9n, co,o,0)=0(1,3,o)+oco,1;1). cc

=

fdt

~ ~ ~d..ot > l~ >, trte>, (<'), (~)

Ar.e.

Cl), (.R.), ) c-p), t;z)

4. (C)

-itt.~ 2..C,~J +0.8.y_3

LI; LD.

AAt.

LD

-

Section 9.9 Section 9.9

139

(G)

!:t:(2>5,l,·3)=

= ~~ 1 +0~z.+f-~3 +~~4 =~~l+5~3~~~4·

a•

Jt ~ ~~~~Ml. •

5.

" •

p

LI ( Nv-~,-wt.~~ ~ ~

~) 4Md.~ o.nt ~ (2.') !o ~ fl...r l 0 -2. -2. 4

d. Hf.v..c(. (~~.s.s-> ~OJI(

•

~~ ~

41 ~,Ao~ crvJl.

~~ ~

1 0 -2. -2 4

~ ~ ·~~ ~~ fRP. GA.ts, M .u..o, c..M

(2.5):

I 0 -2. -2. Lf-

0 l 0 0 3-+ 0 I 003_.01 0 03 2 O l l .. 3 o O 5 5 -ll o o 5' 5" -II

~ol4 =-l7/c,,ol 3 =1~/30)
=

o o 5 -I &, o o 5 -1 ' o o O-b 11 ~.. (o.) y.lA, ~ ( 1- o.,..J. ~ ~) ~~ =-Ii:.. Cb) Y~

.,. ~ ~ .s~ ~-~ ,~tk ~ ~ t~ k.~~~cz. ~~ ~ J U' .J: ~ 4. LI Mt (ML'l'kC}.S.1) ~ ~.~=~ ~ i a,*o. tt

8.

(b) 2..

(c)

9.

(b) 2.

(c) 2.

3

ce)

2..

(e) 2.

C-5-) 3 (f) 2..

c')

2.

C-h)

2.

(') I

Ct)

3

11. Lit~ ~(11.1)fdli4~~~-k.=s.

Section 9.10 142

Section 9 .10 N

1.

N

ug n"' = ~cc~ - ~i)'Z. + n~nt. - t~I M

dl\~ll"th~ck =2.(C~-Ol1t,) =0

~ Ct= a*=±}.·~{?_ . ./

Section 9.10

143

Section l 0.2 144 CHAPTER 10

Section 10.2 I.

8= (H)' !?=(5i), ~= (t), ~:: (-1,2) 2}<. l

2 x2.

3)( 2.

l X2..

8~= (~-~\(~1) =(~~i), ~8 ~M ~, 8~=(~~)(~) =(-~) > ib~not~J I 10} 5 I' 2><13~2. l 10 34 Bx=(5"0 1Y~)=( 17)

""-

A

IO

J

~B~J~, M6=(-1,21(b'i)= ~

!?t= (g-ng-i) =(2g-~),

-o-

1><2

2'1<.2.

(-5,s),

1~2.

Ai.~~MC~, ""'

z~ Mo-t ~) ~= (~}c-1,1> =(:H), ~z= r-1,2>(~) =2.

Section 10.2 145

Section 10.2 146 ~~ ~ I;) (4- 3 \ + ffe; (~ I

A= (o °'

bXo b) =(o oa.c) (o cbyo oa.c.) =(ooo) =Q, A3='i·'°' o~ oo oo.. clb ec1oo oa.. ctb c) e =(o o o o oe-1-1' ctf 1) , Pi :..:t:.: (oo ooooo.d.f) o , A=e.

~ 0 0 c 0 0A.. c, ""000000

A'Z.._ (0

~-

0 0 0 000

Cl 0 0 0 000000

)

0 0

0 00

AO

3

0 0

T

0000

00 0 -.f 0000

0 0 0 00 0

0 0

0 0

q..

0 00 0 000 0

,..,

0

"J

000

0

N

-

Section 10.2 147\ 1G>. (d)

.)H'" . ;

~ (J:~')(p

e) == (t-8n>d!·b) a I.-A + C!-t-H)A)A - ,.., - - = I-f\ +IA t(c-nAXA) --

_,..

= I-A+A - At..

,..

=--:r.:Ai..- =i-I. = ~·.;

--

- -

-

.

-

Section 10.2 148 f2

(

2. 3

xa.. d.,1' ) :

( b) A ..... ~ -: o C c

(2A:+3C. \ Ol

0

2b-t-3d.) 0

f31

i

~, ~ = (-~~ -}~) (C)

~= (~ ~)

(ci)

@=

(-~C(

-!f?')

2.ll..-t 3C

A<>

:o

=

= n_

2.

--' _

~b + 3tl. 0 ..0-0 C tt."1..0: :: - 3 ol.) ~ · I ~ Q.::: ct ·

~tid.=a.ntr.=-~@'~',oob=(3

Section 10.3 149

f = (2.)

1I

(

/2.

o o

Al

0 0 I 0 0 0 0 \ ( I 0 0 0 0 \ /'l 1o o Yz. o Yz. o o ) = Y2. Y4 o Y4 o ) '12.. o /z. O o Y2. o Yz. O Y4- o V2.. o Y4 o Ve. o Yt- o o 112. o Yz. 0 ~4 o t/4 1/2.

0 1

0

o

00001

pl3)

OOOOJ

00001

I0

= pC2.) p<•> = •/z. •/4 0O 0Y4 OAIOOOO) o llz_ o Yi. o o =

(o

V40Y2.o~

IV,...

'V

Y4 o Y4 '12.

0000 I

oYz.oYt.O

o o 112.. o

1/z.

00001

(I()

0 ~Vq.o 114•14 1/8 o V~ O 5/8 0

0

000

O

5/8 0 V4 O YB I

rp~)~~ ik1 A~.b.Jtn4"~3~eo.,~~-a21~f;~ =114) II

~"

2.4.
II

••

"

"

rrkt 'fr)+. ~~

••

"

II

" ..

"

II

''

"

"

"

••

..

..

"

••

~(~~):

A:=

~ (EC2.,-tJ,[3,oJ,[1}tJJ)j

B: = ~ ( [[5,3)2.SJ) [2,0.1,-" J] ); C: = ~ ([[9,1,-1],[2>0,1],[o,4,"J1); ~ ( (fS& &• B)"3 + StC&w C); ~~~ [

2'tl2.4.0 3b,, ~. 5

JS83C,.5

l lOS'zt.35

'7201'7.0

1'1&, 33. 5' S-453.10

102501.S 1'014.5'

l

Section 10.3

7. (e)

11~ a,.t)J ---·J

(232.l ~i) = tf(i ;_i)~+t[(Z32.~J)-(~; 1)] l 3-2.~4)-+(~ l 74 I 1 l '74 l = ( ''7 '15" ·35) + (0 -I 0l 2.) I 1

· S' 3

I

-2. -l 0

:li3 "

-bl3)

r'TI

=1/8

J

~1 "~C3) :5/8. ,-2.l

Section· 10.4 150

8.

Fo1.. n=2,

(b)

(tl.n Aa2. x:t, \ )(a.n-X-1 +a.,2. 'X.z. ) a.,:_ a.u. \x2. J :: ( 'X,, %i.. a:-L~, +ltn ~, : an ?G~+ a1t. ~ 1 Xz. + a,1 'Xi,Xz, + °'n ?l~ =Qn X~+2~ 1 , 'X,' 'X-2..+ Q.u. 'X.~

:'fA

~ "'~ == (X,: Xi.)

't~-t{j a.u=i,a.u=-3, (C)

f~~ ~~Cb)

Mrt

= ?l"f- - 3 X.t + '"Ix.

a,z.=3 (~c.),/).08=(V3)·

2..

~,fat rt=3, tW'

(: au ?l~-t t\u_X~+ a33X~ + 2.~12. x,x, + 2a..,! -x.,x.3+ 2..lta.3'X--i. 'X., ::: 4~f t "~ -'<:~ + 8%.,X.2. + 3X,1X.3-2.~'Xt3 3 ~ au=Lf,a.2z.=l,a33=-1, a.,2.=4, ~.'3= 3/'1..) ll.23=-1 '°O ~=(4i -f1 ) ~TB

312-l -I

Section 10.4 .

alruj1t?t~17~ ~l=

2.. (b)

-G,

~ 31\J. ~

2.(IS)-C-3X1i)+0(2S)

.2t = C-'1)(-C>)-(1)(4)+ S"(ll) =87 v

~ 3.rul ~it= 0(25)-2(-IC,)+5(11) (~)

-G, I

~I= -j -"~~1 5~ \-= -\b~1 ~q. \ =-\ 0~ _t0 -~I =-(1)(-n){ ~) =8'7 0 2S l'1 87

S'

2 0 I0 0 3 J -I

0 4

5.<~>T~

= 8'1v

cLt = a.f (he-eel)

4. Cb) ] f-~ (~)

= 81

I 5

s0

1 2. 3 '

¥

Tf

=_ 00I 42.3 s3I -I"0 2. o 1

~~

o

I 2. 3 '

=_ O 3

l

-l

1-L _

0 4 5' 0 o-lf-s-12

l 2.. 3 ' l 2 '3 ' 0 3 l -I __ o 3 l -1 'f vllx 1\ 0 0 ~ ~ 0 0 ')3 4/3 =-(I 3.\"3-. -l2.; o o-•Jj-~ :: 132. 0 0 0 -t2.

3

.waltt~): . A:=~ (t[~,o,1,oJ,[o,3,l,-1] [o,t+,5,0] 1 [1,2,3,"]1); \ clt:tfA); ~ 132, a..o. ~ed.~ 4(}>· J

Section 10.4 151

a..

0 0 0

o '= (tl)J~~J(f)::: ll(l>e-ccl)f o ci e o 0 0 0 f o b c

rp~

Cb)

= -i-+2e-4-e+i = e..

Pn =Ncn) =n~ Qn.

'V

fo't ~

en~

~ n-+o0.

v

~ rr~tM.t~ Ml( ~ 2n{n-1) ~; -fc't ~ ~ Awf. ~ . 2.(n-1Xn-2) > ~ Aa ~. 1

'1'~) to ~~ Nf1'l)

~

.!

~ oJ..o fort.

'll =I

hl-Ll
2.nln-1)+ 2.(n-1Xn-2)+···+ o ~ (n't.-l) ""' 2.~3 o..o, n-+ oO..

* R~ tk ~~ ·

J.l.Jt

.

n+l

v ·

.

=l+-X,+~2-+···-+ ~n-1+x,n ~ -LlAo. -fort~ 'X.=t:I ~Pu<- I- 'X,MAt. i' 11 ~ >~ n...L_ -?t tk 0/0 . 'T'~/:J/M. ' l-X,

~ t>.Mtl ~ ~ ~=I

Jri:h ~

1

(AAAti. tk ~ i'H~''° ~ fo'lth.t ~~ ~) ~ ~ nv.w..,tt n(d·-1)/s =0+2:1+ .. ·+Cn-1Xn-:z.)+n{n-1) ~~ ~~t4 MlNT. _ · (/V\

NOTE: This application of difference equations, studied in Chap. 6, might be suitable for discussion in class. The result is striking, that the number of calculations should grow asymptotically proportional to the transcendental number e.

Section 10.5 154 Section 10.5 l.(b) st=l, ~=2.) tLI~:::l>ftLI.~=t (C) J7.. = 2. , ~ 0, ,. : 2., =2.. r&.) .n.:: l, " = 2. > " =I , " :. I

=

2..

=I, =I,

(f)

n.=2.>

"

C~) (.1t.)

5?.=2.)

"

JL= I)

II

(~)

J't

=3)

11

=2)

&i

= o,

(-k.,) (t)

J7_:3)

(M)

Sl= 4)

11

=Q

ln)

SL :: 3,

"

== I,

(cl)

II

= l)

II

II

.n ::' .2.)

" ..

II

=2

11

=2.>

··

=2.,

"

=2.

"

=I

=I ) :,

3)

II

= 3) = 2.) '' ::: 4 J =3 ) " ~(~~): I\

J

II

>

" 11

Jh,_ ~ ~DwNl

•t

=2.

=3 = .s = 2.. =4

=3

A: =~(([4,s,o],(3,,,oJl)i ~CA);

~ 1.. 3. C.P) <123~~~)= (1).

cc> (!~X~)= U)· C
(i

st(t\)=IJ rt{~l~)=I, Yl--'l'.=3-1=2.-~~1~

sic~)=2, sic~t~)=2, n-si.=

2:-2=0

.oo

~ gX~i) =(I). Jl!8l= I) si
b;,.., ~

Ao IT\.()

~-j ~e..t

7JY~)= (:). Jl(~)==2, Sl(~~)= 3. Jt(B)t-Jtl~lf) ~; ~ A .. ,, . c~' (§ g~ Y~~)= (:), JHB>= 2, 5lf~l~)=3. " o o Ax. c;) (~ ~ ~ ~ \(~~) =(:). su~)=3, n
Cf) (:. I

'+ ~

2.

I 0

AO fY\-0

'X..3

I

3

1

I )

Xiq.

4. No. Fdt ~~, (1 o) ~d. (t I)~~~~ ~,~.v.rt ~ ~N~~ ~1't~·~·

s. (~ ~ -; ~)~(~ ~~3~ )-(i ~-~ ~)~ (! ~ ~ :)

2.

0 1------

3

0 0

J.------''

·. -k o o o I ------.. 0 0

0000

m,

0 0 0

0

...

%*

~ -R~m.*T~-t-4 ~~.k1if..t ~

a.o,fn f4~ ~ (13) ~ E~' ~ n=" 1\. .. . c~.,~2,···,zn~J = X'X•••

X

x x

n-l 2.

0 () ... t

.: .: ... 0.

0 I

:

I 0 ... 0

z. I

~~-~ct.,. .. ,~n--l ,, •• •• " ...

..

:!

..

••

••

(+1

cln-k

%11.

ocl

~ k= 3. T~

.:

..

l'-. co.)

~.

oz.-

H~+ 2.0H = o H2.-t-~l5z.-~o =o

H+OH-H2.-0 =o

=o

H2.- 2H

H,_ 02. OH Hz.O H O dl.,

Oa-20 =o

'W.t

O

O O O O I -1

-l -1 0 I 0 100 0-20 0 I 0 0 0 -2. I

Y2

0

~ d.o ~ ~~ ~ ~) ~ )it°M6 ~ ~,fo't ~: ~(~): J A:= ~ c'to,1,-2,0, o. oJ,D, 1/2.,o ,-1,~, 01,r-1,o, 1,o, 1,-11, (1,cf,0,0,-2,0 J, [0,1,0,0,0,-2.Jl)) ~(A);

~

I 0 0 0 -2. 0 0 I 0 0 0 -2 o o I 0 -l -I

ao, ~(~), H2. I I 0

H2 ~2H

H2 -2H =o r:1t> &z. -2(J =o

O o o I -2 -1 000000 (b)

l -2.

I

0

l

0

02. "¢ 20

~H-H-{j=o

OH~H+cr

HJ'-2\-\-B:o

Hi'~ 2H+O

ct~ Hcr CQ H I -2. 0 0 I -1 0 -l 0

cit>

0 -2. -2.. 0

I 0 0 0 :-2. ~

o

I 0-2 0 o O I -I -I 0 0 0 0 0

AO

,..:.e,~4 LI~

H2. ~ 2H . ct2.~ 2ct Hct~ e.t+H w~3

LI::.a~

Section 10.5 158 (cj s~~~=

~2 cH 3 I l 0 -I

-I I

0 l

o

o

o

o

o

-t

o

10

0

1 I

H2. \::J

0

O

O

O

0

-I

O

O

0

o

o

-I -2. -I o o o I -J

-I

0000

~i-l...:o ~ ~ 4~ ~

MS"""

1

.

1'7. S.u.k

rrw

~H

C(J

CH 300' CH4 CH305H

I O -I -2. -l o 0 l 0 -I 0 0 0 -l I 0 0 l O 0 -I -2. -2. I 0 0 C) 0 l -l -2. -l 0 l

~

oo

000000000

t):L+ c~, ~ OOHH2.+CJ CH3 tOH ~ CH4 +cr

CHiX1 +OH-:::. ce +2.l-\2.-'" 2LJ . CH300H ~ CO' -t 2112. -t- c:r ~

.

Rcl v

~lL~~»~ ~ ~ 1t4 fo1M>)

b

~ ~ ~f~ 4,b,c)e,f,~,.i,~ ~tk:t

err

(L)tt(L)i.(L'f1tfML3 )e(Mr1•f'yt(L'l'-,.);f-CdCMLT°i.)} = M MJ M: e +f 0 A b c. e f 0

+a ::

L: ~+b+c.-3e-f+l+.t+~ =O rp: -c -:f - 2! -2~ =0

e.

e 1 ~ .i.

f)A ~ AD.

0 •

~ .iv }

ell, [ 1 1 ' -3 -1 1 l 1 0 0 l 0 l 2. 0 2 OOOltOO 1

-rt,-ot,,.,

1= ol.,

~

l

.i..= ctz. >-It= o13 , cf= oi e = c =-ctq.-2~3 - 2ot,, b=~, l\. = -ot,-ol2. - o<3 + c/4 -3(« 1 +~4) ~ Ol 4 +2Q3 +2ct1-ct5 , M I -

H~

I

Cl.

-2.

b

0 -2.

c e f ~

. h

~

::: ol, -I 0 0 0

l

.., i I

+

-I

0

0

0

01

-2..

-(

o{~ 0

+ol3

0

0

0

0 l 0

I

+

ol.q. -l I 0

0

0

0

0

+«s

-1 I 0 0

0 0 0 0

1

Section 10.6 159 Section 10.6 cLt A

"' = -2,

(\-l

~ ~

I

(2. -'f-1 =(.3h_1 -'512. 2... )

= =l -3 5

,.,~: ' ~"-'A- = (-13/2. -S/21' 2. Vs'+) (1 o) v 3 2. = o I ·

Section 10.6 160

r-

A

:t

Ii. ( b)

I 00\ (I0 O 0}-(0 0 0 I 0 - 3 0 0

(

-3 ' 0 ) 2. 1 I

0 0 I

..2 -7 0

.40(1 .. A)-l :T+B-t-A'l. ,..,

(cl)

.

=:;:

l'J

-

;...

I ( 0 3 2. I

0

0 I 0 b I

I

-7

A'2, =

0 0

~

0 g) f\3 - O 0 0 I) Q - - ) A-z. =(O~IO

oo) +(o3 0o0o) + (o0 0o00) = ( 3I 0I 0) =(I010 0 . oO -2. 0 O -23 -1 I

o o o) =r. - (000018 4 0

1 -4

Section 10.6 161

0 -3 0 0 -2.-1 -'1 o

.

,.,.

-21

0

ioooo) 0 0 0 0 -12. 0 0 0 -4 18 0 0

)

A3 = IV

(oooo) 3 0 C> 0 0 0 0 12. 0 0 0

0

A,,.= 0

, ....

- )

f

(0

IO 0I O) ()3 oo O )+ ~o-l2.0000 0o Oo ) + oO 0ooo C> 00) = 4I 0 0I 0o0 ) (I.-A)'=I-+A+A'+A3= 0100 + (04000 1,0 "" -..; - - ,._ o o O o .. o 0 0 0 ·12. -3 (

.40

000 1

.

12.(b)F~tkHt>JT..lN\~(4):

-2-1-, 0

-4 18 0 0

12. 0 0 0

"

1'7

1

- ,

'

(34-2.0 o o )-1=I(3o-2.0 o o ~, o o ) -. = {' o 0 -,(~ o o) -210 o-~o oo 2

10 0 2.

5' 0 I

1

-210 5" 0 l

1

o 0

~

l )-1=I+A= (l 0) •oo) =I - (ooorfi Ae=O o { ~ -SO ' 5' o o o (2. 00 x!-'3 0y2. 0 ) =( 1/3 -112. 0 ) 3 0 0 )•I = 0 ( 4 -2. o -5 0 oo -&lg o Y2.

Naur:

I

2. 0 0

-

•

•5

I 0

1

0 2.

10

13.

-2. l 0

0 ..

0 ~

1

-

2/3

- '

0 0 f:}()-210

So I

,.,.

0 2.l

#V

I

0

~ =5-1£ :: ( ·~~ ·~~ ·~ )-·(~ ) = (-ic,-~ ~~ y~ )=(~~2. \ M> ~A.ct~.

(t:l)

1/3 V4

Vs

2.

so -1so 1so A2-

30)

~tt ~11'~ IW\~'~) ~

(b)

~=(.~ :~ ··~)-1(~)= (~;J;,~' -,~~ :1~;~2.x~ )=(~~~.1) ~ ~ ~ .'33 .'25 .2

25'5:S"b -134,.2 12.'"· B

2.

l&.tfl I/~~~ Qi~·

=

,,.8

2.

21~ 0 /11..,.,s =0.00032a

(1tt "lk

fi ~..;.... <13.n)

..I

(C)

14.

X::

""

(.~ :~i~~) (l) ;: (-i;~~J -;l~.i~ !r~~~ \(~) : (-:.~~~l k, ~ .wL_ :m :is .2 2. '33.28 ~1%.~ 1,S".'7'7 A2 30:,2.) ..tn.t ~...;;, ~

( ff 1)""1 ...,

J

K =I. PnX-~~ lru 1

-

,.J

..,I

8 ~ C6(A-1)-1 f 8-I fl ==A~~) -cA· r-=A...., "' - 1 1

1

1

)

~.

(C.)

(

~8

AO

[

Jx~~ \= (-~ID )•

2. 2.

'X.3 )

Lln:: 2.

.u,, =5' ..U.13::

I

=

.Q,_, .ull 2.

.4cr

111 Mti. +.Uu.::

S

121=1 AO

..Uzt

=3

=

121 Ltr3 + Lt2.1 = O MJ .Uz3 -r 13( J..tll = s .Ao J'!.\ 4 ..231 l.t,2. t .Q!2. .Uu. 2. 132. =_,

=

=

"'°

1.3, .t..l13+ 1.i2. ..u.2s+ Ll33 = 2.

rrk k~ ~ ~ ~

Ao

.U33

=- 8

(4l _,z ~1 A v1~) =L~) ,~ F '<'·=o, "a2-=-7,) ,., 3=-32.. "j3 ~ 10

Section 10. 7 164

Section 10.8 165

Section 10.8

Section 10.8 166 !b) (c)

fra~+f~)-otfC~)-[Jfr~)~ (J.i~~+t.';~J-Ol (~~1;.l,.)-~(~~~a)::: Q', ~ ~· +~N', ){olMi +~Ar-2' )) (M, Uz.) (f\f& Nz.) ot.u3+@>rV3 -()(. ~3 -fl tJ"3 JJ 1M.z.+ olt'(.U 1'1f2. .+Mz.IJ'l) + ~t.,V. N"L - otill.Ui -P>N'il\1'"2.\ oW3+@>Nj-
f (ol~+~~)- olfr~) -~E<~) == o<."' =( (

- (

-

(a.U,

~)

0

~~JAO~~.

Cd.) L~ (e)

Cf)

N~~

fCo1.g+~ff)-olff~)·f'fl!}

(.u +1) =(ot.u,+~tJ;-4-1) otilz.+~N'2.+l -ct. 1

A(f\J""+1) ('-"'. . ~)

U.t.+l - \ ru-2.+1

::

1-o<-~

;:CJ,

~~,PO~· 3.(tl) ~ ~ ~~V Mrt ~ ~~ ~= ci,~+···+
Cb)

~( 't) :: A~ = ~ ~ A=(101 ••o. , ) . ("'.• xn) #'V

,.,

~-

,...,

-

f=I. #\J

-

1t

(?? f )(~) =(~). (ff H~ )-.(fit i:c~IJ~(g~ l ~:~1c3 +c, ) l 2 3

3

t 2. 3 C4

C4

0 0 3 Cz.

0 0 0 Cq.-2c -ct+C1

F~~ ~-~~fdvwtJ.Nf. Aet.~ ~R:: .ncA):: 3

3

~K=O ~ £=Qr~tk~~~~=~ ~v = ~fR :: 3 ----{) \) 3

E~ ~ ~ ~ ~R=3 ~ ~W=~fR'i°==4

E~

N-to~""- ~ ~~=z ~ ~ ~~ ~- -fdl ~- £ ~ RJ M

4. ~ ~ - ~ ~ ~ ~ O\.t-=l-0-~ -,,.X ~ ~ ~. ~=~% 1fkt cJ..- ~ ~ ~j i-M. nC~)=3, ,._ ~~ R .o ~

i.t- Ab

~ ~ K~ ~

1 fi: £(2,0,1, If> C:l,0,1,2.)"I', (t,3,1,3 )TJ.

kk

~ 4:A<)~~~~-

~. (0.) M:!:2, n: 2..

rr'o .k tM.t·fo ..
"'i'2. N;" N'"i. tJi rv; rt;_'l.

NJ ·Ni Ni (\f2. 8.,

(ct)

A IM~ ""

Section 10.8 170

cc.)

oo2.][

F(x) - [001 0I ,..

N

-

=

oIo o-1 al[J ·l/4'i Y,ft va o

0 I l ...3 0001

o 0o 0 0t -l 0 0 -I 0 0001

0 I 0 0 1 0 0 0 000 l

T

B~

8~

•/,.fi 0

0 0

1x

0~ ][ :1

0

I 0

0 0

l

-

Er:

[-~:~~-~HI= [~~3 1

=i,fi,,..__ f~, Mt{~~~~ (0,0,.ri,1),~ R R RX -....."--.. " .a~:=1,~=1,Lic.=-3, ~~~f~a! ""x ... ..... '\ P ('X1 1j 1~,I )- (I,-aI, '12-3, I). ~

t~

~

--.
\

\ J /

I

I

RRX

-""!:"!"'~-c"-1

-Ji (ct)

0 0 .S,'I{, ~01 00~ ] [o:l ~1,87 o0 j-[.~5"53 o 0l ~2.~5'5 o 0o ][•S2.53 -.5"'% ·S253 _ [ or 0t 0o 3I ][ o' .-,so o o 1 1 o .1-,s1 .~ao o .2.~SS" o .'~553 o o o

F(x _ _ r) -

0001

0

0

O

I

o

o

O

t

0

0

l

Section 10.8 171

f(Xhe ) = N

A l'\J

[?] 0 I

(\ ·~·~I ~

F~

rll.S3~41 3.'1'751 1:32?5 I

=ll (O,I, I):T-(0,1 0)~ \\ =\\ (0 0,l)l\ =.1. 1

1

T

= n (l.2.Lf3~ ,3J:i85,,2.2.' 3'7f- (r.S3,~, 371'7S'7, \:32'75 f I\

= ,( (\.243,-\.S3,tt)i.+ (3.S85,-3.~75'7)i.+ (2.2&,31-1.'32'75)-z. ~I ;l()t4 ~a\-~~~. v'

Not J-0 ~:

r, p

= o.~~a>

Section 11.2 172 CHAPTER 11

Section 11.2 J.

~ :: Wll I a-'>-1; I =l'i>.

(Q.)

1!A \ :: (HS·-q.

=0

r "-::

-1, 3.

) (o) ~ ~ . (01 -2.0 o0) MJ ~~=Ol, ?la= ;.a' (e-3f )~ =(-2.I -2.4 ytx., A~z.1 = 0 A. =-l: Ao e, =
A1 :: g:

0

·

At

cl:>) ~ ~ x.o.:> =b• e ' , dct)

· • =ti.eA2.t ~ (•Sf ~

(A,-l)

t' e/'

1

-4 ~2. e

>-.

2

t

=o


e>.,t - C>-2-1)i e"r.t =o © ~ i\., =t: l..:z. t4v. e>-,t ~ e>--zt o.rJ LI .b-0 CD ~ (A,-•)t•::: o ~ t:z. =o CJ.Mel ®. ~ \i =0 ~ (Az-l ) i'-:. 0 'P~, ~=tz. =o ~ W<. ~ ~ -tR.e, ~ ~ 'Xf:l:)=O, "3r*)=O. q,

2. NOTE: This problem makes an important point, one that is worth emphasizing in class in connection with the Markov population example.

Section 11.2 173

-i-A ; I:: o o 4'-A

Cit) 1 2~>-.

Section 11.2 174

/\.:: 2~-s, 4. ~ (g _;; g)_,,(gb ~ g)_,?<3=0,X2.=0,~,=oc.

(2:-A.)(-s-A.'14 ... A.')=o

Ai=~: (8-2.t):f=Q

0020

!t'•=

/J()

(g): ot{g ),

y~

0000

l

~fat~~~~ (g).

ll.

1 l ' O) (7 I ' 0) ('l lo O)

I) ~--r -O · ( n Az= ... 5: ( JJ+5....,

_ _

f>. 0 0 3 0 - + 001o-t0010 ..ltX3-0,'X.z.-r' o o Ho ooo o oo0 O X, 1: 1'7

f'

~ f~=rr)=f(T), ~ ~ffl~~~~ (~),~· cB-Lft )z =~ ~ (-;-~ ;

=

A3 4=

g).. ,. x.3=9i,

-X,2.=

'&13,

x. =1t ~

0 0 0 0

~ ~3 =c~~') =¥('~'),°'~tot~~ ~(D,~· 4-A Lt- 't 4 4-A 4

. (~)

I

"2. • :it:..=XC12-!l) ~A.=0,0,\2..

>-,= ~: \a~~i)~ =Q ~1=ri.(!))

A-0

Az=A 3 =o:

4 4 -8 0

0.

-2 I t 0

0 3 -3 0 }

~ -fo-t~ ~ ~ (\).

0 0 0

0

X\-:: C(.

(8-ot)~=Q ~ (HH)~(g&gg)-tx.3=~,~=0',%,.:-~-'t

w1

~ = (-~j~) =~mt'im, ~ka
·L (-1::)

r (-i!at 6)-.Jt(:-~~ g)...,. (~~3-i g\-+(b ~:~ g)~ ~32~~

l

'2.

•

(~-3t)~=Q'F

A\=3:

\

:-5-~~~~;~~gf~g-5~g~ 'X-,=a.,

~· =ol(!) d~-frn-~ ~ ~ (\). A1::A3=0 gggg) -+ ?l3= (!, ~=
J

= ..., {

/>()

°'~flt .w-U ~~ ~ { (.2)

-~_cs.\ I

l

2!o't~

l=-A(A+l)(A-2)=0rA=01 -l,2.

l-A

A,= 2.: (8- 2 t)~=Q. ~ I)

AO

rn m1.

(-2.. o l o) (1 l-1 0) ( l l -l 0) (l l -1 o) ?..t-'1 g ~ ~i~ ~ g -} g£-'1 g ~ ~-; ~ g

~ 1 =ol ( i, a.~ fdt ~~~

( l )

~ ·

'X.3 --'1

=tt.>

x~=ct, IC.1: Ol

~. (b)

(£-29f

. ['I }:t =~ ~ -I t 2.l l 2. OJ 0 ~ [l0 2.121 3 3 OJ 0 ~ [112.lO] o 2. 3 3 0 -+ ~4= Ol, X.3=P>1 0 2.

3 3 0 4 0

I 3 5

3 3 0

0 0 0 0 0

~

0 2. 3 3 0

0 000 0

=(-30l-3f> )/';;.,

IV

-

0 2.

J..~ ,...,,--c::t2. 2. I

Section 11.2 176

NOTE: This problem gives a nice graphical feeling for eigenvectors and eigenvalues, and you might wish to discuss it in class at the beginning of the discussion of the eigenvalue problem. Even if the space is more than 3-dimensional, the figures hold in a schematic sense.

Section 11.2

fe't )N(. ~ Jl~-2 Jl.?.:: 0 > /"t$ Jl.: O, O, +Ji, -.J2. . ~ ~ ~ o ~ ~ ~ ~ 1>-0 ~ ~ xct) =A~ot: ...., Be-Ii~: Cent o.r.-cl ~~ ¥i. 1)(*=) • \J.(, ~ -.R.o.ul ~ dk ~:I.~ - ~ otJun. ~ 1 ~ .J: M ,.,:);, +k ~ ext - ~ ~ -tW-

o.,yJ_ "aC"t) : e'nt MJl,

1

0

xlt) = CA+'Dt)e t+8eat4(f)

u~ ~ (-~

ce-ur

fJ.X\i) = JL(~)

OM"'

o(t)=~-

~a~~ ~~ c~ A.~;i)

A.,=o,

~·=Ol(! );

.\z=1,

~?. =(2>(~);

A3

=2, ~s= '1!(7)

~ (~¥i)= Ol(\)eot+~(~)et+¥(T)e2t,~iotk~~.

180

(c)

l ;..k~ l = .2. Ai.- ">.. ~ 3 =0

l ~-A~ l = ~t''

r8-~~)~=e ~ =

cct) u~ ~ "-1:

,

~ A. :: ( 3±'5 )12. • 1 A,= (3+~)12.) ~. = 0( (- ~.a)

Az. =(3-4!)12.., ~2. = @(-l~.J!)

if, ~ 1 =ci (:); "-1.=-":3: I, ~ =f{b)H (~)

{~)

X.' = x,+ 2~

'd'= 3X.+4'}) rr~,

Section 11.3

183

Section 11. 3 1. (b)

I8-'-~ I =1 ~:>.. o~.>.. I = ,:-.z).-t :: o, 7t.,= l+./2.1 ~. =~(J-1

2

CLn ~tm& ~

1 !Rz. X

ol=~=1, ~, ~w-+4

(c)

L\ =l-./2:,fz.=f>(-J.-1)

. ( } _ • ~~l..w~ ,.~ t(,d_,),(-.n:_ )1 ~

1

l~-At I = \~"-} -~' = ->..(X-t) ~ 'A,=o, ~·= ~ (~ LAz= r, ~\ =r(bL A3=-t, ~3 = lf(i) ~ ~· ~ f fR 3 ~ f (fl. (i), H)J 1a-"f1 =1r· "~>.. "~>..! =->-3+•2).t.- 3G.,\ ""3'2. =.:.(>--ax >..-2J = o -i-~ ~ g) (1- -t g)-+ (b ~3 -~ g)+ A=8: ( 2. -4 o ~ o o -3 o g6:to 0) g Mx2.=~, x.=~) llN\

ce)

1

2

1 2 -2 1 l

(I l

X3= C(,

3

°'

~' =

(:)

~L=A.3=2= a~ H)~(H gg)-. ~!~(&~,~.=-~-~, ~=r~ti= tsC?)+«W 10µ~~~~i4~~~,bt f z = (°?) a,,..J. Lt ~3 =(!>( ~) + {b) ~ +kt f'z. •r3 = ¥ + ~ =o . W:

e=1~11=-2,~: ~ ~3

= (-:1.). it..o.,

~- ~+n-~3

m,m,rn f1JN

.

(~~~~ 1 ~) ~ fo't (f)J {~),(L),(i> cl'll ~ ~: (2.) (-2.) 2 C:f) A. =-'3, ~1 =0(,(-~); .>.. =A3 =~.. !JC= (3 b H ? . Lit ~z. =(h) ~ ~.3 =f (!) H C4°) .-eh tkt ~z.. ~3 S~-4'1( =0. L.J: e-:it AMtl. ~=S,---a, 1

2

::

AO

~3=(-~)-~,~~.~~

m,m.(1J.

1

Section 11.3

~,=~;.=OJ ~=

(i)

:fl)+f!'m·

(4 akc,,

k.~ ~~ ~ f.4~ ~·

~ ~,= fb) ~ ~~=(!) A3=~4=,, ~= ({(*)+~m1 ~~~~ (~)AM
~ ~.~~ l~i)t)i(~),(:}

Tk,ik Cil

~,= ~;.= A

3

=OJ

~=ol(~ )+~\i)H(!); · ~4 =2, ~4 =b

m

Ao~ ~ ~· k6 ~ (~ H~).(!H~) · 2..(b) A,=4,

~.=DG(1L "-2=-2., ~,_=f ("1).

i =

No,~~ ( G)

3.

A. 1:: A2.: 0, ~

(Cl.) !11.b>-.

=

d~>.I

~ d-(1,...}; ~'1M. ~· ko.4

fot !Rz.

ct ( I )

A.= ~+d. ±,J«t+<1.t- 4
=X--
184

Section 11.3 185 4. F~~ ~ ~ :tc> E~.t,~ 3(CL) wt. ~ it-.o:t ?.. ~ k 1- ~~ 2 /l_ =cl ~d. b = 0 J .w.. ~ C44(. 8= (~ ~ ), ~ ?..,= A.z. =Q. D.M4. u ~ = (~) = ~(6)-t ~ l~L ~ ~ 2-~~~.

1\

'°· I

f)

(A-0,

tk

=

II

I

+ 2.~ 1 - X.:z. 0 j X.1Co) = X 10 , X 1fO): 'X.~ 0 II I ~2. - X 1 + 2.tx,2. = 0 ) XafO) = %20 > ~2.(0) ~~o

Ke· V>'~

~,

=

~- ~ ~ 'X.~ .u.. ) x, £0) 'X.10

=

=

M..': -2.~ t + X.2.) 'X..~

M(O)

= 'X{

x:z. co) =~20

= rv;

rv '= x,-2. 'X.1.;

0

x2 rr-1um~~~ 1'~ 3.~~1tW tk ~ c~~~~, :to .wlv4 ~~ ~~) ~
-eo
/. (b)

= Ol~(t+ cP1)+ ~~(,f3 t+ cp2)

Xi.Ct)= cl~ Ct+ cl>,)-~ A:z.) 'X.,Co): I= ~ ~~ =o = «.~4', -(2>~4>2. ©

x:rol=O:: 2. X~{O) =-3 = c(er.>cf'z,

=

(!)+@ 9 l Q.ol ~4>1 ®+®~ -3 =Q..ct~¢J

cf>l

(!)

® ©

©-@ =9 I = '2.~ ~Q:>z. (1) ®-®~ 3= 2~~C£.>cf>2. ® 'f'~ > @-z.+©i. ~ 4oc-z.::: lO ~ cl= if0/2.

® =}

®

=l.5'8l *

=~ ( /2.
@-z.+®~=9 4~-z.=10, ~=~/2.~1.sst~ 1 _ _ _{i) =* cpz.. ~ 1 (1/2f>)_:: ~ 0.~l"2.::: 0.32.\7 ~ rp-lu_ ~ ~~ ~ :wJ.Q. YLct ~ ~~ ~.

*

=

Section 11.3 186 1~>

~,(t)

=l.58L~v,(;t+0.321''1) + 1.59l~(43t+0.32\'7)

Xa.(t)'=

(c)

u

~ ~ ( h),

%1(0)

"

= 0 = !X~
?l2.(0)::; 0

= o<. ~
~ '°4M
=o = ol Cf.,)ti>, +.[3 eca:>4>z.

~

AO

.

~ ~~~·:~

(a.)

5 = ol Cr.:>cp1 -

~ B ~£t>z.

>
-t

f.

= "

Fo;- ~)

+k

x," =-x,- (x,-.x2.)

l-

> - --Y2113

2f.: ~q,2.:: 0 ~ 2.{3 eoctiz. =-s't'z. - o' ~ X/t): ~'t ~1{3:t X2 (t)

8.

~: (0) :;l~ (0) :

s:

'

~ il..c.t

«.,>X:1,>X...3

=-2x. +x.z.

.-,k ik

,_,.1 ~ ~

1

~ = + ( x.-x2.) - (~z.-X-3 ) = ~. - 2.~ + ~.3

(b)

xf = cx.cx.3)- ~3 = x2.- 2.~3 ~ s~} x~=~~{CJt+4') 1 ~ (~~~i~ Yt\=~(K\ 'fn~

0

r

~ ~

I

+d{~}

'{,!)

"-1= 2-.(2 =0.5"8C.) ~.=c<.(1,12.,1)-r, w,=../X,= 0.1C.S" ·A.~= 2, ~2.. ~ (J,o,-t c.J2. =,.r>:z. = I-414 A.3 2.+'2 3.'tl4, ~3 = ~ (1,-a, l )': LJ!r = ~ :: 1.·94g Ao ~/t): O{ ~(O.Tf,5 t+1) + ~~(l.414t+cf:>2 )+ ¥ ~(1.848t+ 4'3 )

=

=

=

'X,2.(t)= ~c{~(

X.3 tt)=
(d.)

II

·•

)

+ 0(3 ~(

_)- ~~(

iN. ~ f~ ~ ~ 1ra ~·, ~1., ~~. ' X. 1(0)=1

)1:

It

"

GJ,,"';i.,t.l3

)-Ji (J ~ ( )+CS~(

ti

"

)

)J

~~ ~ ~ ~

=Ol~1 + f&~cP2.+ ~~~

'X.io):O= '12~~4>1 -.[2.'lf ~cP3 'XviO)=O= Ol~d>1 -f?'"*"cP~ +Cs' ~cP.3 ~:co)=o= o.'7,S'«Cr.>¢1 + r.lf1tt~C4:>cf>2. + 1.B481 - \.9'f8~ '(/ CDcP3 ?C,3(0) =o 0.'1,_5Cl C(.)d>1 - 1_.41~~ <¥>4>z + t. 848 ~ eo4>3

=

.

•

.

~ ~~ ~ ol,fl,1',t\>,,d>~,cf>3, M~ Ant.~~~ ~ ~ ~~ o(.Al.Md>. > f~3, Ol.ct:>cf>•> PC/:)4>2.,
T-L._

M't,

hJV(.

t.1tn
de<.>~.= 0

kot 3~- ~~Cf>~= ~ coct>3

0

=o

"2J ~ 43

=t/4

Section 11.3 187

q,I : 42.. ~ ¢3 = 'IT/2- ) ot. =l/+)

rr~v.e )

f?i ::

\~

l/2) ~ = 1/4

A(J

•

X1 (:t) ::: ~ ~(0~'5 t +1T/z.) + .Y2. ~ ( lAl4t+1T/2.) + V4 ~(l.948::t+1\"/2.) o. 25 Cr.> o.7,5t + o.s Cd:) \.4l4t + o.25C{:) 1.848±

=

Xi Ct) 2

= ,rz.;4 ~(0.7,5:t+1T"/2)- tJi/4 ~(l. 848t ~1T/2.) =o. '35Lf-C<.:> o.ri,5:.t -o.35Lf- e<.:>0.1,~.t

X3Ct) :::; 0. 2.5'4:>0.7,St -o.Sep1.~'tt

10.

~ =(:11 ;:\), 12.~j>. ii~x]

="7-42M 440 =o A, 1 =22.

,4()

X(t)=

"'

+ o.25'ep l.846t ( ,)

(CJ,= (22.), ~ 1 : cX- -I j

"-2. =20 ( cJ:l. =,/20), ~~ =~ 0)> (cx.,ct) )- oc.(~Cfflt+ ct>,))+ 8 (~(.Jrot+cP?.)) Xit) -~ ( " ) \- ~ (>/2o t + cp2.)

-

t=.Lgr( 4 O.ll

.

Section 11.3 188

l Ao ~oI r2.-1 ~-2.A "r-JO) 3 -S" tl ~

0 -1 ( l 3 lf...14 3

I

o -I

l

-3

5

-I J

~CS') _,~~ ~~ (-10) -~11 J ~ ~ ~(q.). - c'ttcc4-> = ~ •~

=-t.~8

Section 11.3 190 e2.. ~ ' "·2.. - .: 1. M. µ~~ a.. ~~>-:u~. G.v..t o.. jl~ ~ ~ ~~ ~ AO tW . ,--tk ~~-~I~);;;, ~L -.--..~-. ~--.H ,, ' . -- ~~() dJ~f~ (1,0~0). ~) ~~ ~ - -n-- ----.. ~c!1 = a, (7)-K ~· + a2. (-2 )* ~-z.-+ a3 <1) ~ 3 ~a.I J..,~·· ~ "'-~ ),.,.t ~ ~- 'fk... ~C-it) _...,;il ~ Q.z(-2)-l;:~Z. ~ ~ ~ia0 .J:, dk.;.t ~ tk (7)~ ~ MnU. ~ a, 17)* :to ~~) ~~ x,0'.} ~ ,k ~ k>, ~ ~~, ~··

*

(~) NOTE:. 'r~ 6"' ~ ~ Mt°~· N~) ~~~a:! ~ ~ ~ ~ 8 ~ 4 LI~~~'~ ~~'.o ~ GI.~ (~ r4: llM. ~ ~) .l>-0 (12.-1) All~- J;.,...f..4,

'fr\+~~~~

~= ( ~ (ti)~ ~.=4,~·=(~ );

Nott;t~+k ~j'/.)

Az=2,

~z=(i ); ~~vo} ~='i )+~~-!)

4. ~· ~ ~.·fi.:/:O. rp~ ~~ ~r-~~j(, ~·· JS.

AS\(

~31~

~4>~

LI J,,x

NOTE: In Exercise 13(g), above, we noted that the power method works if the eigenvectors of A are LI (thus, the method can work even if A is not symmetric), so that we can expand the initial vector as in (12.7). Likewise, for the eigenvector expansion method to work we merely need the eigenvectors of A to be LI so that they provide a basis; then we can indeed expand x and c in terms of them, as we've done in Section 11.3.2. The only difference is that the ci 's cannot be calculated conveniently by the formula ci = (c·ei)/ (ei ·ei) since that formula is for orthogonal bases; we need to use Gauss elimination or other such method. We will need to do that in Exercises 15(d)-(i).

Cd.) B~~-,o:t~~' J,,.:t~~- ~ ~ "-,=4 1 ~,=(1,o,o,1)T; A.z.= 2, ~2. = (1J-1,-1)1)T j A,3 =A.4 =o, ~ = ot(1,o,o,-1) -t-f(o,1,-1,o)T '60 J.N(. ~ ~ ~3 :=- (1,0,0,-1 )T D.IV\a ~ 4 = { 0,1,-1, o)T. Nott.~ ik ~,'° aJ\t. LI. .u M ~· ~ e.·ez. .... "'" :po. ~);toT UP~ -, f =(3,-1,l,O) = c 1~ 1 + c2.~2. -t C3 ~3 + Cq. tt<1~ C1 + C2. + C3 =3

/\:1

- Cz.

+ C
- Cz.

- C4

c 1 + C2. -

C3

=-I

=I

c,= 3/2.) c2. =C, c3 = 3/2., C4 =-I

::: 0

.

J..O ~ ~ ~ ~~,"°'' ~ (33) ~ ~~

""x. =

31

2..

(-f)

e + 0 ~2.'Q-T~3-0-i-4=2.~·-~~3 3/t.. 1 e J.. e 3 + 1. ~A ~cs.,~= -J.

4-=T~·

>0

.l..

IO

•

Section 11.3 (e)

~

~

A.'.c. a.,y.J,

"-.,::'+

~'A>

:I}

-C2. C 1+ Ci.

-c3

5'"2 ~ M.uJ.

c~ ~t. +

C3 =-1, Cq.

=3

=1 •

f

c, =o.

.

.

.

S~ Ca =t:O ~ J.6,, ~d ~ NU)~.

(li.. ),

~ ce>, ,t,j --tk ~ C1+ C2. + C3 : 2 -c2. + C4 = o c1 V2., c2. -•12., -c2. -cq. =I c3 =2,c4 =-l/2-. 6~c 1 *0~~)~j~

ruw-.(.

=

C1 +Cz.

<~)

~

c! ~3 + c,. ~ 4

c, =.3, c2. =-1)

+ c"" = 2 - Cq. = 0

- c2.

~=4 ~

CAM.

~ ~ llL)

M))

~ C1+C2.+C3

S

e~ 'j-~...:...CMff -pantc.t). ~ ~ :10

£- =(I) 2' 0, 3) = cI ~I -+

-

(f)

'

191

=

=-2.

-c3

CU),~~.

T-k ~ A=o ~ ~ 'A.3 =A-t1-=o t:>o, ~ ~ tuq, ~~kt. M.Q~ C3~ Cf~ ~Jnth~> ~~ 2.-~~ 1~ ~~~1fM· ~)~~

1:

£:

O't 1

c, +C2. +C3

(1,3 1 3,I )'::

=I

-Cz.

+cq. =3

-C2..

-Cq. = 3

c1 +Cz -c3

c, ~1 +··· + C4~4

c1= Lh c2. =-3) C3

=O, C4=0

=I

1

.

AO,~ :to (35), AN<.~ tk 2.-~ ~ ~ ..1_ e 1 - ,..3 e + Ol e3 + Be#f_ = e 1- 3 e .. -+ o{e.3 + ~e4 (o1.,r:J ~) ""~ = 4-0 ..- 0 -i. l- _ -r ,.., 2. .... -...... '"" \ ~~ d

=(~~)+Cl(~)+~(~)

Section 11.3 192

!,. NOTE:

The point made in this exercise is an important one, and one which is not confined to this single example, namely, that it is generally true that the basis that is most convenient, in a given application, is the one that is generated by the matrix (or, in other cases, differential) operator that is contained within the given problem. Other examples of this "principle," within this text, are the following: investigation of the stability of the equilibrium population vector in Example 4 of Section 11.2; proof of convergence of the power method in Exercise 12; the eigenfunction expansion of u(x,t) and F(x,t) in Exercise 17 of Section 18.3; and the derivation of the stability condition r < 1/2 for the explicit finitedifference solution of the diffusion equation in Exercise 13 of Section 18.6. You may wish to emphasize this point if any of these examples are discussed in class.

17. (ct)

Section 11.4 193

Section 11.4

Section 11.4 194

'X.Ct)) ~Ct)= ( ~(t)

..,

= Qz =

( 1 4 \{Ci CC>.f2 :t+ C2. ~~ t ) -1 l AC3cr.J,A3't+C4~13t

'X.Ct)= c,~'12t+ c2.~(i:t +4C3~,/3t +4C4~"3t

~(t)= -C 1 Cr.>1°2t-c 2 ~~t+ C 3 ~St+ Cq.~f3t

Section 11.4 195

( _.L_J..)

"2.'

a

F'2. "-'

Section 11.4 196

=*2. (~en+ x )?. + (:Ji +i;1 y -1) -c ~ +x.)t -
~(;tz+x,yi·+(t+~y-

tz

-c Y.(2'. + )~ cY,.rz + ~sP ~

"' .«.2. (H+ f?: x.+ ! + ':! - ')

?(,

-

~;j-: + t£~ + t ~it~

=-k2.( 1- o+ ,Ji(~-t~>r.!t. ) c- ch+ x. )~ - (;h + '(1) aJ ""' *2. (1- [1-1 cx.+'~P J) l .. J ""' "*2. ;Ji cx.+':P(-ti t-H) = -!~ x.1~ (i-t,S) F3

A

A

= ~3 ( ~ x,'2.+ (\+~)'" -1) -~.L- C1+~)j ~ ~t.+ ( \+ "() )~

"'

"-' 1~A I- ( 1+2~ t'-)(- x.t-O+"j)j)

"'

d't)

x'.' 'd ,== -_~i x.% _- ~~~~

,

<11..

2..42.)

CW\cl A.z.%-l.53, ~2..~ ( l

~ Qi''= AQx

dL

A

%: , =_~ wk..n.t. 4o ~ ~=

X.'' = Q1AQ x. = (;\'

Q

_~

0

3/8 -5'/8

t

~

-

v§)

Q

.. =

(-.LfIl4

A,::: - o.q 10, ~,

2..'f2.) I

=(..o~'Lf) ·, ,

.

AO

.

- -- o >-2. A":l ) . """'-1:J f.<>- ~ ~''=A.,i rlL X.''+o.47oX. = o A
FAN.~) _

rw

-- -

~

"W

-3/s) k e=(-111s _

r

('X.(t))- ,. , _ (-.Lfl4J

~(t) = ~rt) - ~~ -

-

2.42. x~l er,,O."S't + ~2.~0."B"t) I C3 Ctr.:>1.24:t + Cq. /:Wv\. l.24t

-.

-

-

-0.414C 1 CfJ0.'8"t-o.414C2.~o.,sc,t+2.LJ2C3 ~1.24t +2.Lf2.Cq.~1.24t ~ct)~ E, en~.'B':t + c2.~o.'s't + E3 ~1.2ttt + E4 ~1.2£4t. To N.M.~ ~ ~ ~ ~ ~ M ~ ~-~p~.d:- ~ %Ct)=

1'

z<* i = ( ;~;) "

(-

0

t )(c 4

,Cf:> o.c.

~t + Ez ~ o. '-8'. t) +(2 f2 )cc3 Cr.>1.24t +c4 J:i<.;,•.1.2'1t).

Section· 11..4 197

dn.1 ;,.,,_

~j; o,.,.i;l C'Ju,.l'LUL f1!fl;w-~ J

a,,,._

~ct)= (;~m = cs- (- 0 ; 14 )~
.Th Law-y~~ N'\nlt..

'~

'

~~

I

HU;f rt\~

G. ~~~-~.::to_~ o..a- Q= (~.,- .. ,%.i) ik, L;) ~ ~ /W'-~ ~~, ~~IW 8Q= ~li,, ... ,1.~) = C~i,>···,ftin) '7. E~: J{r Q""'AQ =D, ~ ~{J"'f\G =91» AQ= QD, .......... ....

i-

......

AQQ-• =QDQ-1

,._

A.

,....

..........

,.,,,,,,,

,

A= QDQ-~

4-0

rr~ A =QDQ- QDQ.., =QDI])a-· ~ -aD~~-· 83 = 88.?_ ~ ~1x~r1 9p:z.9-i = Qmb~ct = G~?Q\-rlMC! A-0 o'Yt-. 1

2

'f~

Q:= ~([[-2,-l,1],[t,O,l],[0,1,l]])j P:= ~(Q);

G[' = Ii' L -

~)

(Ab

D"° = -

11° ( 00

*-,• -i -1)~ -1 2.

(

00

0 0 ) i 1000 0 1000 0 5

~~ -(dL 5 1000 •

=l {

0

0 \ . 0 0

0 a.. ) ~ J:- ~

0 )

F,v;~ , . = ~ C[[1,0,0), (0,1,0), [o,o,Gt]]),;

R: ~(Q~~R&•P)_;

I N(.

,. _ . - - ~ -

+ -r_

'"'9'~

PiJ

J.AAl

rt

a..,,

Section 11.4 198

~, +-~~~'14, (3+0.. o A1000 =.!. - i-+ a..

-2.+ia..

4- -I+ a.. -2.+ 2.a..

N

-l+'l.)

2.-+ 2 a.. -l+d.

3+a..

~~; ~ ~ ~ .2_lCOO( II 22.. lI ) . 4-

- ":

-4~

',4

~-~-·.

--··

. ·-·

. ~

-z.

l 2. l

AO

..

,, ~ _: -

J.. ;c_x_ :

127
J "jj =i21 er/1z.,

. . . - --

·,

-------~~ - -

~~~:: Co2.0'/3.

z.

1- iv.-J. ~ 0 Jd =(Ig) dt,),, ~~~~)

10. (b) WjA A,)~· ~...... ~

•If :12. g 0~~~~1 O g ( 0

(

~+

1 -3 3

o0 ...I, -I2. o0 0 l -l o

0 I -3 3

)

(I0-2.)

I\ •

•

(10.1)) (lo.'#)

~

•

0

I) { o -l 2. o I ) ( 0 -l 2. O I ) 0 -+ 0 0 I 0 I _..,. 0 0 I 0 I O O 0 I 0 I 0 0 -l 3 I

o

Section 11.4 199

O 0 -1 3 I

~(

o o ooo

·(~:) XtCtl _ (.2 0 -l 3 x;rt) ~4

O -t 2. O I oO l O I o O O S 2.

J~ -

X - I Xlf - 2./3 > 3 - J X~: I, X 1 -: ~

o o o oo

-I2. o 0 o 1 1 o 0 I -3 5

ix

1

X.t ) ~3 • X4

(l0.t2.)

Section 11.5 200

6

Ck.~~(10.2.) ~~~4 LI~~~~~ ~~. N~,14 ~ ~=P~ ~ Cto.12.) toi4 ~ ~~,"J~)-fc'k .

~.ct)= (C4 + t c2.tz.+ c3 t) ert ~2. (t)= (C3 + Cz.:t + 2C.z) e2t ~it)= (c3 + c2.t + c2.) e2 r X-4 rt) = ( fc2.+ ~C 3 + ~ c2.t) ear+

~

c1 er;

Section 11.5 201

Section 11.5 202

ti

+ Ao

"

~ ~,M.

= C3e:t- C,.e 3 t + 2t +{et- e3 t)h. ~(;:) = C3et + C4 e3 t -1+ t +
'i.lt)

~
hit {C,-tCi.)hECJ (- c,-+ c2. )h, =c4

Section 11.5 203 at)

tit)= (c3 +-£)et-{cq.-t-t) e 3 t +2t

~Ct)= (C3 + 1: )et+ (C 4 +t) e3X + t-1

5" o)
:: Cs-e~+Ce,e 3x+2.t

=Cs-et - c, e:tt+.:t-1

5 e .A. = ( eo o) 1

eA . =I+ A+ t, A+ ... =r. + (o4 • "" ,.., I 1.

I

oo oo o)o + (oo oooo oo) + J. (oo oo oo o)o + o,.a. =·( o ooo) o

3 0 O 2. I O

.1..

2.1 12. 0 0 0 • ' 3 o 0

31 0 0 O O . 12. O 0 O

...,

4t

1

1 3

I 0

!i2. 12. I l

Section 11.6 204 Section 11. 6 1. (b)

8=(f? \)

(C)

5= Y'2. Vz.) (

l

I

•1a o o

Q- -

0 I 1/,/2. 0 0 0

(o

l

o

~~· ~: "-"~=,,2.,3.~

Section 11.6 206

''

'

''

.....

I -

'~"j=-3X,

Section 11.6 207

,._

x, = X-1+ ~2. CC)

f

=~r. .

= ('X.~+4X 1 X.:z. +4X~) - 4X.~ + x,:z.x.3 = (?(.·~ ;~s:. Lff~~ -t~!;:.3 +tit?(.~)+ ft<~- i ".3 )t. = ~J 4 'X.z. + rl; X.3

~

(cl.)

~2. = ~.3- ~ x., i_, = ~l lf x.,x.t. ... :ic.2. x, 3

-

Section 11. 6 208:

Section 11.6 209

0

( ·~..e.-r ~1 fc£A~)--'I ), ~\...

~

y- L

~

('2..

\0

= ~1=1:L

lV) £1) "~ =~\;

ott, -I -;_

ea.=(.\). 'PW ~, i.r<. ~ -tk ~ ~

i\. 1 1, A-2=3, ~ «,=-1=0 ~ P\.=lcn3;

i\=l ~

P=1eL

fA..M.J..

A-3~

~~~~~~AO PCft,::-kL/3) ~ ~ ~ . . ~ ~ +4 ~ ~ (")= ct(~1))

P= 1!.L/3.

~ dtdnw.. •

.

~ ~ -t4

•'(1 ~

...,;..~ ~ ~ ?,:...-t4 ~ ~- "'f~~~AAl.-t 0..0.

~

P.,

~~"VP

A,O,

I

Mr{

'X.-+O AANt

~-.o.

NOTE: You might consider expanding on Example 4 in class. For example, the physical significance of the eigenvectors is not discussed in the text in Example 4, nor is the final note, above, that relates the potential approach to the force equilibrium approach.

i

Section 12.2 210 CHAPTER 12

Section 12.2

Section 12.2 211 ~~ ~ ~.=cA.)1,0), e2..=C2.,:z..t,1), f 3 = Cl,.i,-4). (el) .u.. = (2+q.i, -3L 2.)= (2.+4.<.X...i.)+ ~3.LXl)+(2.:xo) e + (2+qA.)(2.)+C-~l)(-2.i.)+<2X1) ~

'

'

CLX-.L)+(1:(1)+(0YO)

+

....,,

(2.){2.)+(tl.)C-1l)+C1X1)

(2+4'.X1)+ (... 3,t)(... i)+ C~(-4),

e

C1Y1)+C.i.'J(-l)+(-4)(-4)

= (b)

cc> (cl.) (e)

e- t + ~ e -9 - ~

4-5".i.

* 2.

11=(o,o,1) = ~~,+

~~-:

1s ~3

C3-t.fi IB

.... 3

e

- 3

•

=(l 1 1) = t-.\, e + e + ~ 18 e ~= (L>2.l,31) = t-4-21. e + 4+5.l e + 2-11~ e JS

.u. ""

, )

s-2...t

~ _,

2.

-

~

_t.

1

~ .... 2.

3

-3

~= (o,.L,o) = L e - ~ e +-1 e 2. - •

(~) .t.t. = (l-.l., o, o) = -

'

1A. e 1 + .2-2.t. et.+ ~ e 3

1

-

'· ~·~ =

18 -.3

""2.

.....

M1 ir,+···+..un..UY\

'

-

18 ""

j .u,, ...,.u1'\. o.ru. ~ &

= lLl,1'2.+···+1.Unl-z. >o

=0

~ ~ CS"b). N~, (~ -t ~~) • ~ : [ O(,l(I + ~r-J'j ' ... '

-{

.'

llJ'\.l

ol.l 0 ,

-

+ ~/\JY\.} (-tJi'>•• • >,urn.) = (ot.u,+ (3t.s; ) Wt + ... + (~.Un+ ~~) Mrn. = ot ( .u,MS-, + ... + ..U.nM-ri) + ~ (Ml'Wj + ··· ~ ~ W-"')

= 7. u.Q,~ ~

at(~·~) -t

tk

d,U n.

(S(!:!9· ~)

Mr~~~­

c~+ol~) • c~+a.~n ~o u~nz. +oL !r·~ ;.~·~

+ + Ol«- n~ni. ~ o l\ .U.1( 2- + Ol rr.:;::r + Ci M • l\T' + lCX \'2. l\ /\j it '2. ~ 0 Lit 01. =a.+:h D-Mtl. ~~ = c+:cf. ~ . . n"'.......q\t. + (lt+ibxc-icl)-+ (Q.-,.ib)(c+A.d)-+ 1ct1t nl\jnt. ~o ..... 11~1\-z. + 2.o..c+ 2.hl + (0:+ b'l.) \\~I\~~ O S.it ahtt 2c+2tt H~l\'2. = 0 ~ ~ ~ = -c/U~l\2. SAt ~hb 2cl+~l\~l1'2.= o ~ b -
=

rcJ.

=

=

M>~~~' !. :JS1:. 2A.'2. c'Z..+ cl'l. 'Z. 11-\6 11 - II ~II?. - n~nt. + II !l!" II"" 11 ~ II ~ 0 J 11~11'2-n~ui..- 2.(c.'2.+ ct..2·) c~-+a'Z. ~ -

+ cc.'"+cl..~)

11.u. u't-n l\rll't. """'

#OJ

1~·~1 ~ n~n n~n

v

~o,

O

e

.... z.

Section 12.3 212

Section 12.3

ce> ff'~ m...t ~ ~

l

3. Cb) J8-~"J; l = l+fA

MB=o. (T-kt 3>'41. J\4W"~o.. ~ 1ik1At.)

2=.i.~>J = A.?..- 3A + (3-2.i) =0

.AO A.= (3± ,/-3+ g A., )/2.

Section 12.3 213

e :

/:JC

_.I

s~~,

Tk

Ol

(2.l-J+~-3+8..(. l )T 2. )

i\.2.=


ez: ~(2..t-1-2.~)

~ ~ ~

rn.ot H~~

j,l

I)

+r>-<> ~ -.f'!OI)

Ao;{.e. "-,~ ~ ~ .k ~ (~, ~

~ ~) ~~ ~ ~,'°' ~ ~ .l.tt ~"'4 (~,~ ~ ~ ~ ~1 .. ~2. =0(~(0.115-2.~'~.t):t:o). FAM~ th..~~~~ ~(,.nt(-3+8•.t)); ~ l 0 bb4~ + 2.'f02.5..t.J AO. . ~ i\. 1 z ~.'3'32.-t1.201.t > ~· ::::-c:i(0.'332.+2.2.0IA., l) ~ ~ 0.,,,8-1.201.A..,

4. (c) rr~ ~ ~

~t.~ ~(-l.3'32.-0.2.0ll.,l)T

~(~~): A:.=~ (tC 3, t+I ], t 1-I ,2..J] )j ~(A)j

r

~, =*, ~, =c1) t- -t :x: )T T A. 2.=I , e2. =(-t - t I, l ) ~~ ~ ~-- ~ r\~~ ~)~ ~) ~ /\.>,Q.,a/l.f.~ ~ ~. •fz. =
=

(rJ..)

A.t=l, ~1=(1,0,0{ A.2. =2, ~2 (4+.i.; t, 1-.i. ~ 3 = l+i, ~.3 =. (I, j,,/3 > 0 )T i.o Mt.. H~~ MJ~ "->;;:, ~ 4~ ~ ~£ti4 ~'.o ~ ~ ~ ~
=

l

6

~f-.ct, ~~~A-''° AA.t ~, A.No.&.~ ~,'° ce) >-. 1 =0, ~ 1 =(2.-2..i.,o,1f

OJU.

4

~~

ce.d.,

~.·r3=1).

=(-

i\. 2. =- 3 > ~ 2l/3 , I , - ~ + ~ l )T A..3= 3, ~3:(.l/3Jl, i--f-A.)T

~

N:,

H~ ~) ~ ...t.M~, ~ ~,/.)

~~:

AAL

.nuJ. ~ +k ~~ ~

f!,•f?.. =(2-2.l)(i-~+O+(l)(-i-~l) = O ./ ~' • ~3 =(2-2i)(- ~) +o + (1) ( ~ +; l) =o v' ~2.. ~3

=(-.i/3)(-i/3)+ I +(-j4-~lX~ + ~ l.) =0 v

Section 12.3 214 A1=-l, ~1=(-1,1f j Az.=2,~z.=(2>1? · ~ = (-A.,J) · T ) A. =2-A-) · ~2.:{..t,I) · T Cb) A1=2-TJ.., 2 1 cc) A..1 cs+m )/2, ~ 1 1, C3+ffl)/4 )T.; .Az. ~ (s-ffl )/2, ~z = ( i, (3-l-33 )/4 )T (ti) "-• t3+4.l)/2) e,:: (1J cs-q.t)/10 )-ri 1\.2.= (3-q-.i.)/2, ~2. 1, cs+q.t)/10),.

S. Ctt)

=(

=

=

""'

=(

Section 12.3 215

NOTE: You might consider discussing exercises 13, or 14 and 15 in class.

14. 8~= !2.· \J~ ~ ~ ~tk:t ~·% =O.W.Ji, c· r: = (A'X.,)· ~ = ~· (f\-r) = x·o =o ..; ,..__,....

-....-...

"""'

.,,,,,._

~

,._,'W

~~~

fi'"E =(H Xt) =(g) ~ (i~) = o(.(!1)

15. (Cl.) en,

/c,-ci=O.\

J ll-0

~ ~ £:. (~1) =0

~,~~~~.:to~~~~

(i ~ ~~)-i> (~b ~~-c,) ~Mr<.~ c~-c,=o. v

l

~,1d~(~~-~tk~
(f)

I 3 2 J c1 ) l 3 2 t c, 2 -I I 0 Cz. -+- 0 -7-3 -2 Ca-2.C1

)

( l 3 2.. I Ca _,,. 0 -1 -3 -2. Ca,-2.C1

ff*f: = (t-~ i-LY'!~)=Q .~(~O _;_lO ~o -~o ~) ~ I -3-2.t. -4 -::. ~ ~ . o 2z.:? (30l+3~)/(-2.-l),~ J.A ~ (2-1-.i)ct, ~ 3 = ~2+l)~, ~2.

4=

,c.o

1

0 0 0 0 C3-C1-Cz..

0 -7-3-2. C3""3C 1

3 2. 3 f C3

Ao w<, ~ Cf C -Cz.

:0

../

%=0l,%:3=f, 'I-

.

~)'°° M ~ ~, ~)

=-3~-3(3, %! =(1-l)Ol-(l-t~L) f) s= \(l-i.>_c;i~~r'l3) =ot (~L l+ ~(-~i.L) = r., + ~ 1:2.. 0. (2.+~)~

(2.tA,)ot

rk,

1

Au(

1'1\..1.Ul

2..+~

2+~

-+-

~ not.~~

0

£·~.==o ~ ~·E:t=o;

J.. I 2i-3 Ci ~ I 2.. -4 C3 I 2. -4 c3 2. I I c1 -I 1 -5' c4 .i, t U.-3 c2.

o{

.A.e., (1t)c1-3C2. +cJi)cq.=O I ~ (-l+2l)C 1-3Gz. +(2.-.i )C3 =0 ·I

0 3 _,

Cq.-t-C3

0 3

-<}

2.Cf.~CJ

o ..i.+1

..3.t,-3

c*.t+c2.

~

0 1 -3 0 0

o

0

o o

(Cq. +C3)/3

C4-c3 +c 1

s(c2.+4~)

-(1+.iX°C3+Cq.)

Section 12.3 218 \f'7. C
=

-

-

Section 13.2 219 CHAPTER 13

Section 13.2 1. Cb) el(P, P')

:=

~ (1-o)t. + c-r-4 f + (5"-3)?. + (o-z. )?.

=~

ct(P,P') = 1(1-' )t. + (-~-S)'2. = ~10&, 3. dCP,Po) = ~(4.3 -4.z)2"+C1.1-t)t. =~0.02. =O.lt.ll < Jt> kl f AO~ N. 4. cl. (~fo) = ~ (2-3)?.+ (5"-4)2..+ ('7-S)?. =~ ~ 2H5' < Jt, ~ p ~ ~ N. 5. c:l (P, fo) =~ (.01)4+ (.04)1·+(.05)4 + (.03)1- = ~.OOS\ ~ 0.071 >-'Z.) ~pk MA AM N. Cc)

~. (b) ~~ ~ fl1l.l- -3,S j ~) ~ (C) ~~Mt. X=Oj ~j ~ (ci)~~~X=Oj ~~ ~

(e)~~

~

)l

~

Cm) ~ ~ tV\t. flx,>x. 2 ,x 3 1~ x.7+x.~~2) X-2..=oJ; ~; ~ C<') B~ ~ ~ f cx..,. .. ,x'+')l~ 1 =01 o~x.2.~1 1 o~x3 ~1, o~~~ 1J, {C " )(x. 1 =1, .. '' " JJ

l( " t( f( II

)IO~Xi 1 ~t,x..2.=0JO~X.3 ~1JO~X.4 ~l}> I ,, , x, =I , J) ti

)

)l

II

)

0 ~ x. 2, ~ l ) x J =0 J 0 ~ ~'t ~ l

J)

Section 13.2 220

~

7.

x I ~ I ) 0 s xz. ~ I ) XJ = I ) 0 ~ X.4 ~ I J )

{(x I)' . ') x.4- )\

0!

fC

o~x.~1)o~"z~1)0~~3 ~1, x4 =0], •• •• ,X 4 =lJj~_)~

)\

[(

11

)}

(b)

(a.)

3"-

15+E 15

4+€----1 Lt- - - 4-Ei---

(C)

~1+€~

~~

~l

JS-E

~t-€ /' )f

l

/

·-1, . ) Av<\ \-€ +rwv... I ~ ~ ~(€) = ~4-t€ -2

Cot~).

~~~(E) : IS+E -5: E ( dt 3

3

~~

'\

r

-1

"'

•

AvA:(E-tAw\.\)

&(E)=

~1 (E+~l)- l

(o'l~).

~)

e-~

(f)

(cl) ~2+€

.

/:Nf\2.

'1---...,.....:::=::::...... ~-...,..c;------+-~ /

~2-E -+---___..--L"-'-S-- "

f

2

\_~ 1 (-E+~2.)

x.

.1'

-~(1+€)

~ ~ ~f)=~ (-f+~2.)-2

~~ ~(E)=k(HE)

(dt~).

(dl~~).

1

{!~ ·:,.:

-3

-2

-1

x

2

3

Section 13.2 221 ~.eel) F~~~:

I [fC'X.)+d(X.)]-CA+B)I =I [f(X)-A] + [~CX)-8]1 ~ <

+ J ~cx.1-BI + E =2€ s. €',

Iftx.)-A\ €.

~

.

S~ l[f~-x)+~lx))-(~+6)!< E' J fen ~.i.-.M.t'i ~ 1-X.-ctl < ,rY\AH\f b1> ~ t, it ~ 'tko.:t {9.4) N:> ~ • .., 8

(e)

.

Ll: ~ ftx) =A. 'rk; fo'1. anJ~ ---- -- - ()_ ~ € >O ~ ~ I fCx)- A I< f fJ = ~ c~:f
-~

EI)

c+)

<~l F~ th.t_ W

IH-x.>~c.~)-ABI ~

J

tt. be£)

~U

)

(IAltt)E + IBI€

=€' fol ~

~ E ~tA. ~-fort ~~all€', fen, o.il ~ ~ O
10.

w ffx.)= 'Ix, d
~ LHS = X_,O ~ (.!. -L) = x....:,o ~ o =o X 'X. 1

w-

12. (o..)

x= o

~

t:l=O.

x = (-1 ±00-)/2

=1 )-2..

4- c1yx., ~,

lb)

x= ~

(c)

a.,U ~, ~ i-k. X,';J ~) av\~ ~ ~+~ =±TI'/2.J ± 3W2

(cl) (e)

(f) C~)

~

x.=- ±TI"!~, ±3TJ"/2)±srr12., ....

ill~)~~ x,~ ~' O"'r\t-kt ~ 'd==-3 .. ·: ... .. .. .. .. .. Cu.n.v( d = -x.a ill~, "

"

lvV"y

.•

X,Aj/t. ~'
"

"

~

0<

X~"{j'-+:e_'::I

> •••

Section 13.3 222 Section 13.3

Section 13.4 223

7. ~ f 'ex) .14~ j X0 ~ ffx_) ~ .k ~ Q Xo} ~ tL. ~ £xJXe~~ 0 > ~ f,{X.1o~ ~ r4:. ~ -tf-.:t frx,~) ...0. ~At ('X-r,,~o), M.~~1;) ~~ ~ f(?t,~)=1~:~0 'l'~ :f'l(.co,o)~ c~~o) D.Jt.J. f"Jco1J> ~ca.,.,...lro), 4~ cit" (op).

4

Section 13.4

1 44.

-ai:t f ...a

Section 13.4 224


(Gt!)

fCAA,"-";J) == 'JCx...,_ + 3AXAJ: ,,_.. lX:+ 3l'Vj)= .i\7fc~.·1p AO f ..:0 ~. o-1.. ~ 2.. ~ C •• ) = h(Ai.'X..-z-t ).'L~ ) = 2~A. + .kcx.1+'d'Z.) ~ ~ Ao, ~ ~. -fi( ·· ) = cxl.x.t.-Ax;\.~ )/(2A_x,+~) = A-(x.1:..x,~)/c2'X.+'d) = i\.~('X,;J) MJ {, ~

-re . ~·1~1. ) = >.3-X: e?..X./Cit'\'3) = ~ X:ex1r -,)= X-pc7',·~P AV 1' ..;.,~· 1 ~3. 2

(b) :f ( .U 1 (A.'X. 1 >••• ) A.X.t\) ,. .. >Mm( Ax.,, ... , A.Xn))

= f ( A,,. ..lt 1( 'X-1 >.. • >'X.n) , · • • , A.'b' .U. rt\. (X.a >' .. > ~"')) = (A.'h-)f f ( .lt,Cx., )". ") x. n) .u. m. c~. ,... Ix t\.)) =A\t> f(..tt, )"') .u ~)) f~ ~ ~- f~ x,, ... ,x.n 1 ~ '1,f· )"I. )

M (c)

ll =~ dX.

~

flx,~.:e)- fcx,~,~)

1

'X.,'-~

X4X.

:!icu, AM ,A.t:):: ~ fC"-X', A.~,Al)-f(A.X.,A.~,A.~) d'X

d

x~x.

~x-i\.x.

= ,,__~ ~ :H~',':1,~)- ffx.-,,~) i\. 'X~~ ~ '-x..

= "--k-1 ~

C)X-

)

Section 13.5 225 8. M= fCx.+ct)+ ~(x-ct) .ux.= f'
~.

(c._)

X-";)" + ~·+ ~-zx'd =o. WJ:J.t X=cit, oct

=ci...ttu v'

fcott) ~olt{~ + fccd:)~ + *.?.a.t ~ =0)

~~ . , oc. a.,.,J.. ~j "JlXlt)) =I.Ht), t4'+ .c..l + -kt.«.t. :t-u. =0. ~, ~d o(=l/-k A.tlf'> =AJ~,tt)+BY0
r

"(1lt.)= AJo«i:.x.)+ B'fot-k.'X.) x= cx'.ti3) tkt- = ctf3 t fH c:tt , A-O dl,

Cb)

x'd ''+ 'j' + *.'2x~ = o. ol:t~ ~

J,,

M

o(~t~-l cit o<(?>tf!-1 cJ.:t a

2L 1~ f!:_

ol..2 ~1- t@>-l

ctt

+

aol~;te'-1~ "j + .-\t-z.Md = 0 )

t l-~ r!- M. + _\ :;t'-~c:Lu. + t2.J...L =0) cl:t

~

c:J.f:>

J:. ct•-~.tL' + c•-e- >.f~4' ) +K~..u.' + "*-?..u. =o,

o( ~'t-



;t 3-~M.11 +( 1- ~ + (?>) .tZ""f..U. + -f:.t.ol~t.M. : 0 AO Ait ~:2., ol = l/'4-ft-z. > M) X.: t_~k_?. J t: 2t.{i. tM.''+~ +.t.u =D) MLt} = AJ:
Section 13.5

Section 13.5 226 3. (a_)

> taylor (exp (x), x=O, 5);

1

1

2

1

3

1 + x + - x + - x + - x4 + O(x5 ) 2 6 24

[

[ > with(plots): 'J)~t~~'\~~(~aA. Sl,SZ,53)~ilt. C'.dWVW\WYJ. > implicitplot({y=exp(x),y=l,y=l+x,y=l+x+xA2/2},x=-2 .. 2,y=-3 .. 3,nump ~ o int s = 5 0 0 ) ; rt; tML.. · ~

e

3

: --;; 2-~. ~

)153

(~,"'~Ad-)· Nott: J ~'.t ~ S't~SS'~

Sz.

/1 2

~iht¥.t~

s,

lrt~~;~ clo 54, A."Ni SS' ~ ~

2

e«,

~~1.wt~. > taylor (1/x, x=S, 5);

(cl)

1

1

- - -

[

5

25

(x - 5) + -

1

125

(x - 5 )

2

1

1

3

- -

625

(x - 5 ) + -

4

3125

(x - 5 ) + 0( (x - 5 )5 )

> implicitplot({y=l/x,y=l/5-(1/25)*(x-5)+(1/125)*(x-5)A2-(1/625)*(x5)A3,y=l/5-(1/25)*(x-5)+(1/125)*(x-5)A2-(1/625)*(x-5)A3+(1/3125)*( x-S)A4},x=l .. 9,y=-2 .. 2,numpoints=500);

1

2

3

4

::: x. - CC>Jt

.

7

8

9

I •

?(,

.

6

? x3

lf. fQ.) ~~={~o}+ (~o)x,-(~)x}-3

~

3

l.Awi.~-~l = I~~ x,3 ) < ~o c.s) ::::: 0.0208) ~ l~~-~I< o.02r. Cb)

~?G

=

')C.-

j

+

crqt ?l15 ,

•

Mf

•

f,W,.~ -<~-~)I

~ J~x- (x,- ~) l < 0.0CX'J2' I v 3. Cc)

~X, ='X_- ~ + 3!

.

3

C/:)O

120

;;x_,'S - ~rfr N" SI '· /\...

S"

1~~-
3f

~


=I x,5' I c. s )~::: o. cm 2'04-

sr )J

= J~r~'I ~ (~o.S)(.S)'- = o.oa:>OI04 ~.

'720

v

Section 13.5 227

5. rp~>Jl, ~:

.

~x. == (~2) + ceo2)(~2)-t (- ~ )C'X-2.)?.- + (- ~~) cx.-2)3.

C}Ar
l. 5

~ x ~ 2. .5,

ck>~:

J- Cr.>cf-

3

(?(r2.)

I ~ C. (0.5)3 1"J ~

I CG->j-l I~ I,

W

.wt.

4Nt

2.5

1-5

J.5" ~ x

O"rVl

'·

~

\'fr

}J ~:::1.57

I :;

2 .5.

~(x)= [x.'2.~~) 'X..:t:o o, x=o

'f-l..., -f:
'X :/: 0 ' O./('X.)::

o

x..=,.o

°(jfX.) = ix (X.~ i:) : 2'<.~ k - eot . rp~'

~(2.X.~-b-ep..l) x..::, o "' x

~~~'~ ~~~ ~4~.'/..J:o.'co)~~~J k1N.

--0

~~ ~·- ) t, ~ • I ~'(O) - ~

X:

x~o

d

Yx -O

x,

=

~ ~.L

x~o

?c.

()

=

0 ~ ~.1.. ~ ~J.ul

x.

~-tkt.ic:~-+o. ~I. ~'Co)=O i...t ~~'lxl=tO(...;..-f...c:t...t~......t

~~), ~ O/
Ao,

w.e. ~ Jo'4Lik

~~ j

~=O

~o0

x,

a •

/

Ch) f''to) = ~

et.

~o0

-3

-- ~ *'"'oo 4t et -

~

t~o0

2te

-o

1. • =}Mith

x.~o

-Y~2. A • z. 2e ~ 2:t x,Lf-

8.

'X..n. h~

C}. (b)

'V

~n

~zrrn

Xl'j~

nn

+ G,

en

~

= _L_

=.t-.oo

ei:

±.. -o e1: -

cc> f '"coi = ~ f "c~l- f "fo) = k (~,-{-.. )evic.z._o x~o

1

f(x.)=e· x.'"

(1J.J..'H1'.ti:&_>/.)I~)

_L-tz.=O

- 1/~?..

f
x.~o

~ ~'(O) ~~.

,L...;t-;.-tfu.-~-r ~ ~

~ ~ ~ 'X--:/:0. (4.) f'Co) = ~ e'lx}--o : tk, .:t_z. = ~;t· -x..,o

...tMIV\

~~o

(e-x.)n-+ (OXO) = 0

~2.nn Y\r

~

=..:tc. = 0 , a.a.~.

) A4 n-1t oo

= C3X-2)(-X:-I)+ ll)(-2.)l"j-3)+0){3)(~-+2)4···

=

0

~ ~ ~ _,,~-1)-2{~-3)+3(~+2)=0 ~ (1,3,-2) (c) 3X.'Z..+'d2.+~2.-lb= '{l)(X.-\)+ 2.(3)('d-3)+2.(-2.X~.ot2.)+···= 0 Jl-0 ~ ~ ).,o, 'C.'X.-l)+bl~-3)-4(~+2)=0 ~ 3(?C.-l):3(~-3)-2~~+Z) =O. ~Av(. CAM.~ fu ~ ~ 3X.-+3"j-2~ lbj ~~Pr'\. Cb). A-0

C5f

=

Section 13.5 228

=

10. f<~.~> f[a.+CX-0 a_):t, b+ c~; b)i:-z.J = Fct) F'Ct) = fx x.'c±) + f~ tj'lt) (x0 a..) :f" + 2 c~ 0-b )-i: f"
=

=(~llf:fxx.+2(x..;a.X~ 0-b)tf~+2('dob>f~ + 2(x..;a.X~;b):tf~x + 4(~ob)-z..:t'-f':1'J F"'Ct) =2.CX,,£1.){~ 0-b)f~+ .2l'x;a.X'd;b)ft;ix.+ '4-("j;h~'-:tf"
F '"'* ( )~

ri14wn

4nt ~ ~

~

4(d 0 b)

-·-() f"3";1 T

0

a.a.. -fo.n ~ 2 ~ C'W:Lut.

d

"Lf

8(~ 0-h)

f(x) = Fco)+ F'(o)t +

d~

iy F''lo) ;t2._.. t F'''{o) t 3 + ~ F'"'
= f + [f':lx1 :1: +

i4 [12(~-b)'l.f'fcj] t'f- +···

'l'.Ju.n ~ -j{....t f~: 1

f';jX

> M.

k

'l'S:f { = f(I) = :f(tt,b)+ (x,0 a.):fx.Ctt,b) + C~ 0-b)f"J(~,b) Q.,b + tcx.o-o..f-fx,c, + H""j + t
l l. (b) e~

= e~ + ~e1 cx.-1) + t ~'e"S cx-1)i..+i-"j3 e°1 cx.-1)3 + ... =[ e2.+ e'Z.(~-2.) + ~ e'2..<~-2.)-z.+ -te2.c"c1-2)3 +··· J 2

+[2e. +3e2.<~-2)+ ~ Ct1-)e'2.l~;_2)-z.+··· ](x.-1) + [2.e'Z.+ 4e (~-z.)+···J(x.-1)

= (C)

~ 3XftJ

~ [ t;- se2.. +··· ]c~-1) 3 +··· e2--r 2.e'Z..cx.-1) + e-z.<~-2.)+ 2.e'Z..(x,-1)2.+ 3e'Z..l'X.-l'X~-2.) + t e'2.c~-2)'2. + ~e'2.(*1)3 +4 e2.c~-1)'&.(~-2) + 2e-z.(~1){"(1-2)'2..+ te*'c~-2.)3 + ···

= ~(3r~p + 3'd-Cd:>(3"a)(X.-l)- ~ ~'Z,~(3"<')(X.-I t- i 'J

3

~{3~)(X-f )3 +···

:: [-~3+ 3~3 (~+l)+t~3 (~+1)2.-i C()3 (~+1) 3 +···] + [-3Co3+ ( 3Co3-9~3){'d+l )+ (,~3+ ~ Ci";,3)(~+1)'+··· ]('1--1) + ( f:-~3 -(?J~3+ ¥C<.>3)(~+l) + ···] (X-l)'Z..

- [ t<-1)eo3 +· ·· ]c-x-1) 3 :

-19···

-~3 - 3Cr.>'3 ('X.-1)+ 3Co3 (~+I)+ i-~3 (X.-lf + (3Cf.'3-~~3) (~l)(~+I)

+ i~3 (~+l )2.. + i C/:>3 (X.-1)3 - ("~3 + ~ ~3 )(X-1 )'(~+I) + ('~3+ ¥ £4:> 3 )CX.-l)(d+1 f-

i

ep3 (~+I ) 3 + ···

Section 13.6 230

c~) :fC'l'..,'i)) = Cij-:1> e''Lx,i-+1 ...o. c! ~ ~ o1 tk ~AA.( J . ~ i f x.+ f"j-"}' =-2 x. + "J e'\111;}' = o. •

r

'*

f'd-= "ile~= e o a± r1,1), .oo ~

~' = 2 ~ e-'1 = 2e' ~ O,l)

~" = ~e'J.+ 2"X.;2~· e~+ ~c-'<1·>e'} =
-=>-Ll

'Z..

AO ~lx.)= 1+2e c~-1)+ ~ex-\)

+···

d!b.1. ~ e"1 + x.e"1 'd' + ~' =o

2. (b)

a,,,.-c1.

MJ

"J' = -c-x.+ e-"<1 f 1 ~" =+1 (1- e~''f) ('lG+e"
( 'X.e~+a )( x,+ e:~f

ci+Vj.+3,'t)~'='

JJM. F

;:,(1

":1'= <1+~+3~-z.r•

-2.

~ I/:: -I ( 2 "JI+ '~ "J I ) { I+ 2~ +3 ~ 2.)

=-

eel> JJtVt. ~ ~e't +~'e"A + x.";j e'i ~· =o

Ml

2(1+3'1)

(I+ 2.~ + 3t't) 3

~' = - ~'fi+':P

A-i''=--L+ ~ -~~ a xu+"j) ~z(H~) -i (H~)-i.

= 2~+2"f1Z.+"J.=

3

x,2.(1+~)3

3. Cb)

e f.vte..:l:

rJ./tJ,x ~ e"c} + x.e~~x. -2~~x. = o, "jx.1

r

'-a fWied.: rJ../#

1

= eV(~-x,e"t)

e'*-2~~x~'l - ~i.~&)lx =o)

:r.x.l"J = eVc2e.a.m.~ +i2• ea;,~)

(c) ~Re.l: ~lo.rt.~ e-x.+e"
.

~

e 'X-+ e:c 2:x =O,

l = - e ~-e:

'Zx ~

4. (b) f, = x-.u~~ =o } ~ ~ 13.,.2 x, ..;ax) x2. N.> ~, .u 1 ~ M) A.tz. .t¢ IV, ~. :f2. =~-,u..O.....N"=O S~, f, Wv-.c:I. fz. AA(. C' ~..;... x.,11j,U,N.'4'~. N..vµ, j()f1 /~.u., of,/a.u,j- jafr/~M of1 /arv 1-J-Cf.>N" ~rv J- _2 o.tik 'Of2/~M, of2./d.A.lz. -

af2/c~u afz.l~l\J - -~

-uep(\j

-u- ~o

t~~~.rPlwitkt~~, ~d~ clo~~~ f~ .Ul?l,~) ~ ~{'>'/~1) ~~ ~o-nl

f CX,'J)= (0,2).

3.U.WZ

-1

0

3

=(,uw -".ul\rw2..

t.) M~

e

J.J.!IT

= MW- (w--c,rv) e

Section 13.6 232 8.

(a.)

R~ u

f

F

.

~ ~ ~ dltilx e.u. + x.eu M' - 3~2..tA- x,3AJ! = o AO M.1(K.) (3X~.t..t- eM.)/('XieA..l_x. 3 ) R~ x. a.a.~~ M.. ~ ~&M. ~ x.'eM.-+ x.e.u. - 3x.'2.ix. 'M- x.3 =o Ao x.'
ll/.).

=

1:

W-t

AU~~~~ du cllt ~=I. dJ.J. t:U.t.

9. ~~ ~ (48) ~ ~ (47): (a.)

x= u+ /\i ~= M..-Ar

A<' A.\=

N9

1f

I

(·x.-t ''<1)/a..

=('X.-~)/2..

rr~, *~

I

..U.x ftlx

Cb) x=.u+J\J

d(M,iV) ~(X,~): d{X,·~P ~fl.(11\1"}

Iy2.

M~ 11 x,AA 'Xl\r I = ~ •12.111 l'V'd "(1u ~l\r

~'t ~

-'12.

c4) fdl. .t..tcx,'J)

I

I -1

I=(- ~

)(-2)

~ l\rfx,Ap,

= I ../

A.
Mr(

.wcu.

M J<,

~=M.3-1\1"3 ~ ~ (0.,)) kt Mt(. ~t ~ .U(>c,~) ~ Ar'(Y,~), M ~ nU
~/~x ~ 1= .Ux +/\f"x 0 3.L\.2. .Ltx -31\f'Z.Ar"

=

Section 13.6 233 II. j ~ 1o

~

1 Jt

r ,,_..i.

M~ l\J ~ f~.4 'X.D.M.d. J' ~ -X.~d ~ . s, ~ tStJc..L...,;,,. Ex~ 10. N ~~, ~ c......, ~

~ (10.2).

=u co N" d= 1-l ~N""

(Q,)

-

. N"

=.1l + s = J'l'+ 5'Z.

.

J~ ~(10.Z..) Mrl. ~ Mx.,.U~).-.r-1',l\f"'d. ~ ~ (41t>.,b) ~ (424,b) ~ ~~.l( tllv\d -9-"l\f:

Fdt

~

.u

~

'X.,

fA.ot Llx. ~ ,.,,

=Cf-'_rJ',

=-~f\1 .l..\..

{\j_'d =

Fdt tkt ~ ~~ ~ ~ ~,

Xs

=

-

(¥)(\)

M

(10.2).Mr<.

~ x.Jt,'X.s,'JJt,~s. F~~ ~

~

=lJt+.S)C£:>(Jt?.+s-z.J

~

= (n+ s )~ (Jt-z,+ S

X.Jt: ~ (51~+ 5't.)

/)J,(:J

Ct.:> ( "

"c)n = ~ ( ..

~

=~f\T

M~

-)

M.t.

4

)

- 2Jt {Jl.+ S)~(Jt-z.+ 5'2.)

)- 2.5 { ·• ) ~ l + 2.'l ( •• ) ~ (

)

. ~s = ~~ .. ) + 2s (

Fen,~~ .J ~ ~ ~' ~ ~

··

)

••

)

..

)

.. ) ~ (

(10. 2) A.vt

.

rt\.Lf.d.. AA.n, .Us, N".n

1

N""5 .

F~ t4 ~

.U-'t=I, .U 5 :1 = 2Jt, NS= 2.S.

"'.n.

'P~(I0.2)~ i . '2• l.J-lS:: ( £t?M.N" + ~ 111" X[oo(Jt...+Sz.)-2R..(Jt+s)AW.(Jt...+S...)][~CJt...+s...)

+ 2S(Jl+S)C<.){JL'l..+s·i.')]- [ CO(Jl2.+S')-2S(Jt+5)~(.Yt+sz.>][ ~(Jt'l.+ 5?.)

+2.ii.(Jt+ 5) co (Jt.
'°

cn(Jli.+sX:~
>J) =[ (-2J1.(Jt+s>'.::> ] [ S

s s>(]/u

+2 (Jt+

- [C-2s(n+s)S ][S +2JL(.'t:+s)C ]/.u

=(f:/5+2s{.n+s> cz.-2'l,(5l+s> s-z.-4Jt~_

-%- 2Jt(Jt+S)('

= £Vl\d. l'l.. (IL)

RHS=

(1){2.S)-(1)(2.lt)

f ('p!"f, l\J" ( f ;°f)] =0 . ;}fc)p ~ ff)-t f,....l\r.o=O, r re>Jo'T' ~ f,,, + fl\J" ru-.,,

2(5?..-Jt.42.)/J.J...

= 2(5-Jt)

Ao

=o,

=

2S(Jt+s)Sz +4n.s~ )/.u. 2(5-Jt) -4-

~~~ J

~ =- f~ ~p f\r

A-O

~ =- ~ 01

~"1"'

~~~~~-J~·fdt

(b)

.;L f/\j=tO ~-r

1- f,..,=1- o -

~~

fd"t-p.ud ~ (Au.'t4·~~Ex.~7).

Section 13.6 234 H~,~~~~~~CAA.t: ~~W 1 ~ c12.1).,1Ac.~-tkf ~ f'~ ~~- ~ ~ 'T' .f.v,c
t4 ~ ~ ~~ rv, ~ dN"/op ~ dpt~r.r ~1.:4~ ~~- 5;_;;.~,~~~1 ~ c12.3) ..w.cJ... ~ ~ M ~ f f:-te.l. ~ (c) f)rr, f\J) =(f>+ ~2- )(N'"-b)-RT =o ~ (l2.l)~ a.n.t ~L

-'K1 (12.2.) ~

~AM-

(57) . ./

'l'"jj =

(~-2~X~-2.~)'11 = (~-2~xrrM-2.T/\() = TMM. - 2.rrN"..U. - 2 rrMJ\T + 4 rr"~

M\&

rr~"J = ~ ~ rr = AO · ( {

r

M.Ll

f"f'.ul\T = 'P/\rM.) ~

-2nr rn

4N" e

-2~,,.,

-2J\rfl1

•MM.+ L.IMe

'.uiu- + 2.(1-"')e

(41\J'i.+ eZIU")'T'.MM. + 41\r'f'.Ml\r + 2.(1-1\r)"PM-Tl\T + 'J'nr,v (cl)

-2"1"

-21\1".

'T'l\fl\1" + 'f.u.u. = o

•.u. - e TN"+ e

X.= M.-z.+~\ ~=~+l\i. ~.t.'ll M.U..cl. ~x,Arx 1 L\"(1>

/\rd

=0

1S M4' ~ ~ ~-

1~ ~ I= 2M.Ux -t2~1V-x, 1 . _ 1 _ I 1 ~d .. o = u'k: + l\r1' ~ .Ux- 2.(M-M) > 1 Cu-Ar) ~}~ 1 ~ .. 0 2.U..U.~ + 2Arl\1""c11 . u =...!:£__ , rv.. =_l!_ il/O)~ 1 ~<{ .. I = .U1 + 111"''",1 ~ ~ 111"-.U. ':j- N"-.LI. AO

dl";>X

'dh~

/\T:x. - -

:!

rr~

>

L

~x.

=e.. ~ L- la[ = , ..a. _ • a_ ~ +~ 2(1.l-N") ~ ' .2.(AA.-1\r) ~ ;)X,

i.. ~ ~ + ;)N" ~

]__~ ~

~~ -

;)AA

;)X,

= .JL

a_ _

ru--.u. aM.

_g_ a_ N'-u a"'

1'u =2:).(-111") ( ~ - ~)2c~-rv>(~ - ;nr)T =4 c~-llT) ( ~- ;"') [ C.l.l-1\1") ('T'~-1'"'")] 1

M>

=4(~111") fh-tt~l\r)?. -r~~~»'2.J<'1'~'f'N") + (A.l-IV")l(Tw;2.'fM~ +1'NW)} = 4<~Ar>'1. cr.&.IM.-2.,.AW+ TN'lll" )- 2cu~">3 cr.L\_,..N") '1'.,,,=(N"~M. fu - ~M. ~X ,:M.'l'.M.- !u rrrl")

Section 13.6 236 -

M.

[

l\1"(-1) (r.1-M)"

\

~ ~+

}'f.

Ul\f T, µ.. - (i\f:M)?. Ml\T

+ _&_ (- .u. .,., + ~ "' 1 N9-U, (l\T-.U )?.. •ivl\r-M. l N1\T 1 = ,"1'-M )' ( "'... r'"u + .M.~ rl\1"111" - 2.W\J"TAA"') + ~2.+ /\J"; c'T'M- 'T'IV") IV'-M

F~oJ.t..,

rr-xx~ 1'Ao\"1 = 4ll\J-..u)"l• rL
+ c4.u.'Z.+ ' )'T'N"l\r - c8LW"+ 2.)1""'\,J "J +. 2u-z.+21\r'l.+I (T, -I"') .A.U4.

d d

2.(f\r-.M.)3

(f) x=A.\.'l..+l\r, "1 = v....,_-Ar.

1

it=~c~;-~-J;, c-x:-~12.

I\. rt.

I J

-

A'\~ -

I

21(2.

{"'·:+14 )-~2. d

A..

~ ..tit ..ua. ~ ~

W.e. U. h\.LJd. .Ux, .uAl, 1\1", IVA4

~. ~ ~ J:M•.l:tn......TLv>-,. ~ ~

/WU

A.\

-

I

iM(. '

t I

~1

". -'h

- 2.H ("'+M) "' a -

-'-

- _.L

- 4.u.. '

=

~ J_ ro: - -..L "- 2.. ' ~ - 2.

er~, ~=~~+~~=~~ +i lit a_= dAj

~

R....~+:a....~ :-1~ ~

w ()"i

a__

4t.l )U

.La..

2. ~

rr'XX, =(4h!u+t~~~'f'u+trr"' )= l/u.{- 4~)rr~ + 1k-z-T.uu +~rUl\r 1

rr.~"J = (..!£_ - J.. :a_ x...L-~ - J.. lf.U. ~ 2. ~nr 4M. J.A.

If\ ) 2 1 N" -

+ tu T'"ru- + ~(T'iv-.v..L (- _L )"' + J_ llU 4u-z. l.u. l'Ut.. MM.

r. - SM. ..L

IT' 1Ml\r

- ~ 'r.w\r + 41'N1\J'"

~ =~ ~ +~ ~ Ao

AO

'T'~x.-t T~~ = (.u~ +

1''1i=(.u':i ~ +"'1 ~X.u;'fu-.t)~j'I'"")

= ('r...uu':I+ 'l'.ullT ~) .U'J + 'T'M..u"J'1+ ('l'tnA. .u./T'.-~ )~ +Tnr"G-d

.u; )'fMU+ l..Uxl\T'x+ .U~t\f">
+ (.U'X-x+ .u~'l'.u +

Cl\r-x.-x.+ nr~~ )'l1ru-

)Tl\J"N"

'11-x:x. + llMd.

f

rr1,

= {.u~+.u, )( rr4.1.U.+ rr,.,.111") = o

.ui+AA~*O ~~ x,~ ~~ 1~~ 'fM.t.l+'l'l\Tl\r=O.

Section 13.7 I. Cb)

TSIJ~=• ~

=5+3l'Xrl)4

Ao

o.:t 'X=I

f'{l)=f''t1)= f"'(I) =O,

fciv>(1)

>O,

Ao

f~a..

3

4£~1) , fl.Of k "'-~· ~- f· j ~=I (.fl '11Sfl~=I = I+ 8(-x.-1f+··· ,..., I+ 8(~-1)~ A<> f ~ ~ ~-~· f ...1: ll(.:I (~) 'T'Sf lx:1 = -8(~1) +···) M f ~A. ~.a:t x=I c-l) T.Sfl~=• = 1-cx.-n3 -+···) MJ f ~ ~ ~· ~-~-lit -x.=1. Cd)

2.. lb)

TSflx=i=

3

4(ix.-1) -t···

f = (~-4X.+5)-I > J I = '2.

'V

~

-2.

(2.X--LJ-)(X-4-'X.+S)

=0 ~ oX ~=2.

f''= -2l'X.~4ix.+s)2..-C~'X--4)(-2.)(2'X.-q..){~~tf-'X.+5'f3=-I c.:t ?(=2. .Ao fC'X.) =I - ~ ('X.-1)-z.. +··· , MJ f ~ "- ~ ~ ~=2. • (d.) f = -3lkx.)\ f':: -9(k'X.)-z./% = 0 ~ cit 'X. =I. S~ ,k-x,:: (X-1)-t···,

~ tl...oX f =-3(.f.v...'X.)3 = -3(~-1°')3 +··· ~Cl ~·4· pt.J:-X.=I. (e) f: {~x)4 , f'=4(h't,)3/x =0 ~At X=I. S~ k~=(X.-1)-t···J w< ~At.(.~ f = (~-1)4 +··· Ao f ~ a. ~- tit x.=I. (f) f=e-/:Wn,'X, J f'= e~-x.(-Cl:>X.) =O 4t° nn/2 {1\=±1,:!:3, ... ).at~~ wt

CAlw\

f ''= e~'X- (C/::l-z.X+~'X.,)

=[+ e

1

@ X= Tr/2. +2n1T (n=O,tl,±2.,..) @ x =-rr12 + 2.n rr

. - e c ·· Ao f ~ ~~ ~ X,= -1Tf2.+21lrr ~& ~ ar x.=rr/2+2nrr fa"r, ~ Y1 =o, ±I, ±2, ••. c~) f= x e-x., f'=(~x,-X,~)e'X=o u, ~=0)2. ~) f"= (.2.-2X.)e~ - (2.X.-'X..2.) ex= '(2-4~+x_2..) e_X. =~ 2. o.:t x= 0 (-2e.-z.. AX x. =2. .AO f ~ a. ~ oI x=o ~ ~ a:t X,=2.. (Pt,) f:4.{i-x'", f'=2/../i-2X.=O ~ o.:t X=I. 'f~, :f" = -x.-312.. - 2. = -3) Al() f-/...o...o, a..~. ct~= I.

°'

)

Section 13.7 239 b. Cb) fx =(qx. +i:) ~ (2x.'Z.+x.r:-S"c2.) f~=O

fc= c~-1oc)~p c

.k'l't-_r3.7.4, Cc.)

.

~~=~=O.~~>~f~~~t ~AQ, AO. tt ~ ~·

)=O

B= (f tfo) ~ ,..=0,-10.01,+4.07, ~ 5-k..c...~ ~ ot

~~~tk1~· fx.: 2.X+j + r =0 ~~= x+~+r.=o :ft: 'X.+ 'j + 2~

:o

~ rp.J-- 13.7.4, (e)

..

=01 .

~ 'X.='d=~=o.

( 2. I I ) . f)= : ~ ~ ~~=4,1,IJ~:f~a.~j(O,O,O).

fW' = (2w--x.) .Jl)Cp(w-~t..r~-~~ + ~z.) =O TK. = c-w) ..L1f ( " )= o ~~= <-c)

.Jl)lp (

f~=(-,+2e)~p(

~ 'J'~ 1~.?.4,

8=

) =o

•· ''

~ .W-=x=~=~ =o

)=O

(2. -I ~

0

g~

0) .

0

o

-I

2

-

.

~ ~i\=1+'2:,1+a,1-«,1-n,~AA(.t

~ ~- 1-lbM:t, 1- ~ ~ ~ oS W"='X='d=~=o. Cf) f Ml' = -8.w-+4~+ 2~ =o · • fx= 4W"-8'X. =O ~~=x="j=i:=O

-8'!1

f~::

fc

=

2w-

~ rp~ 13~4 >

=0 -Bl =o

A= ,.._

(-84 -84 0

.2..

cJ.t

20 ) ~A.= -8 -8 -12W7 -353

0 0

0 -8 0 0 0 -8


I

'

'

.

'

.

=

~. fx= Zll'..f'->(lC~";1t.+~~)+':)+-e =O f'd= '(. + 2'd C<.>( " ) + ~ =O f~ ~ + ";1 ~ 2~ Cr.> ( ... ) 0

=

~DJ\.(_

"1 (O,o,o,o).

'7. f ~ =2.'X. + 2.(~-1 )Cd:> ['X.(~-1)] 0 } . . ~~ 10~ =o ~ ~ ~ x.=o,'<'=o,~=I. fc = 4C~-1)+2'LCd°) [~le-I)]= o . ~ 'rlvm. l 3~·'h I\_ ( 2.0 10 0 2. ) 0 . _ ,. II\ . . • ~ - 2. 0 4 ~ ~ - 10, o.v'1q., s.2'1-. LUJ( AA.«. ~) ~'MAM (Le.,~~~) .:t (O,O, I). -./

=

'

=

~ ~

AArt

~ ~~

AO

f

~

i

2~<'it-1t{+·t')=I) ~~~.ti~

~to ~+.,+:. =o. 1fw.o., cJ1 ~ "U.t.~ 1~1~ ~ 'X-'-+':{+ci.=~i:= 1.04'72. .wd:h tkt. ~ 'X:+"l+~=O ~ ~·~

of-i·

.

cM.

Section 13.7 241

Section 13.7 242 (21.x.f)a.

r1t,

t (2I'X.j)b

( 2rixa-)ll:- ...

= 2.!:X~f$

= 2~ f~·

c2.g) b =N

O,_ ~'~ ~, Q.=

bo ,

r
d

12tx.~ f; 2r-x.~ 2.~f1

12 t x ;2.~Xj

2.N

I

21: x.31 2N

~ k ~ (12.2)~ (12.3). Cb) Ella.= 2 r~; > o ./ ~

122.IX~ :rix.~ 2rxf I= 4 [N rx.~ -
]) =

~ ~=ex.,,.

.. , xtJ)

~! t;:[: (1,1, ... ,1)) ~ l~·~I ~U~llll~ll ~ 12 x.~· l ~ ~ lxl ,[N en (tx~ ~ N1:. 'x.~-z..,

y-

MD~o. H~)~ 7.L:.13.'1.s- ~ fdt E°'~>o ~tt l>>o, ~

.~~_To~

1J4

1 Ov.-.J,, ~ { x,=x, =···= x ~ ~; ~ :D>o ~

,

~ ~ 'l>=o. ~:~ c.w.c.. ~

~ J.Nt. ~ ~ ~ -t4 xi~ ~.ct:,~ ~rr~13.1.s~ Ec",b)'A ~~W.~. P~ '~~': LJ:~~~4 N =3 IM-.il. ,}: -{tn ::to N 4.

-f:rc. N =2. ~ N=2: N1)l -z..-(rx..~)-z..=o 3

111 ,

~

~

~ ~~ ~

2xf+2x~- cxf+2xaxz.-+>Ci)=o en.., x~-ZX1 Xz.. + x~ :: 0

ex,-xiJ-z..= o, ..o.o fen N=2. :D=O ~tWx,=x2..

dl..,

N=3= Nrx}·- (r:x.·f=o ~ a 3K'Z..~+3X.z..z. + sx.i.. -(x,z.+ x.%.z.. + x. i.. + 3 1 3 ell, t1t 1

ori,

L.r(_

2.X. 1 ~+2x 1 x.z. + 2.'X.-2.x. 3 )

2~+ 2?(.~ .+ 2.X3°-2X 1X3-2X.1Xi. -2~2.~3 =O

=o

=

(~;--2x.,x2. +Xi)+ (X~-2X 1 X3 -+ ~,'~) +[ X~-2X'2.'X3 + X. 3 )-z. 0 cx 1-x2.)'L + (x,-x. 3 )-z.. + rx'l..- -x3 )'Z.. o , ~ -fen N= 3 D = o

=

~lhc.t ~.=X,=~3rr.k ~)~l>-:to, ~ ~~i4 ~~ ~(12..2.) f. IA.IV'.tl (12.3) AA.t ~- /lo.µ ~ ~ 4J\(. ~fut·. (C) rr'o0c.e.~-tk-o kJ~'l E ~ ~-~ ~~~ ~ n.uct "1o ~. ~ E<11,,bj ~ <12.1)..,,. ~ i~ ~ !'.\) b) />o ti. ~ """""' i.o .d..o 4- ~ ,,..._;,__

-f'j

i

14. (a.) f (,:_,~) =ol. ( ZIT'X.'Z.. + 21T~ "J) =~ 11~~

=V

0

'd=V/1f~z. "'° ~ fl1l,'a)"' o<.(zrrx.z.-1- 2::~-~f) =Fc-x>. 'T'k F'cx) = 2« (2n-x.-¥i.) =o , ~ x = 3 VI zr °""'a i k 'd =Vlrrx."L.= 2 3~V/2n. rp~ ~

Section 13.7 243

II.

I

,

i-....

,.....,..._

'

:.Ji.. I

Section 13.7 245.

I~. Cb) f

=C-t.·2.f-+ ~-i.+(c+lf: ~J

~='X~'J+2~=4

~ ~~' 'X.=~-2~+'4- AC f = (~-2~+2)~+ .'d~+(:+l)'2. = 2~~+ 5~'2.-~'d~ +4~-'! +5 . .

s~, ~

=

?C.

~~,

"3 ~ z.

AJU_

~ ~, A.t:t

~~ q~ - 4 ;! + 4 = 0 fc= lo~-4'j-,=o. . T~ ~ t=•l3,~=-~l3 ~J.~ ~= x-~+2~=4 ~ ~= (~-2.)42.. + ~-z. + Ce+ I)

(c)

~: 2X:j:~=~

~ ~~) "J'~

=~

'd

= 2't.+i!-3 AO (~-2. )i..+ (2~+~-3)2. + £~+1 )2. f"-: 2(X-2.) + 2.{2.)(2.~+ ~ -3) 0 f'r: 2. (2.'X.+~-3) +2{~+ I)=: 0 .

f=

=

.

Tk f-rt -x..=2., ~=-1, ~ --tt-.. ,=2~-~+'! =3

~ ~ :::o.

~'l";)) = '3~t<1-t)-12~'3 = 0 2a+3~3

Mt

X-= 8/3.

(X,

~ry';!-)

=

~ = r-!'}. ?~ '-f{j ~ x, ':j;_2./(s~1 ) ~ @ ~ (1-:;~)z.+ ~ + (I-~~)= I C'l ':1~= ±(1- ~~).

"j-z.+4~-1=0

~

1= (-2.±ffl)/3

.

'j'Z.-t~+ l= 0 " ~ ~ )~ A.vt. ~~F~, ~='-~'J=•-tc-2~-0S) AO z~=i-~(-2;m)='7+,4m.

21.

~' Ex~20 ~~~,,=I w...cl -9'=3. W-'"'1<1\..(.

tU.t.k>~~ ~~- ~~N}(~~~k10~·~

~~i4~1L~~·

Section 13.7 246'

t O'l..

E~1- l~:t ~I~ =o o:ti4A111., ~·~ ~oK·~~ l~:~~~~l:to I~''j ~2'cj 1-:to ~I'll '1~

at tl.t hit. ·

Section 13.7 247.

= 2~- 2l. X. -A.i. = 0 f• = 2tj - 8"-1'd - "''- =0 f}= 22:.- 8~ ~ - A'l. =O 5 = 'll'.. +LJ•t+4ct.=4

f~

1

S~,i..t ~

'3

Cc)

f·=

1

= -x+ 'j + ~ = o

-s: =

>-z. :±2/,[3;

=

1-2A., 'X. 0 -2.')..a'd -A.2. = 0

= f ! : : 2.-2x,~ +>-z. =o ~' =x"+~t.+ ~~=I

s~, 'X.="J ='i: = + •1.f3

~2. .=

.

Ovv.J. x.=~cr= e= - 1/,(3

'd- ~ =0

..

~,J.ir.~MA.t.~~:

~ f= ~,+2'd=~

(~cl~ ~=±1/.(3 of"= - ~ - 4~'L

1

~= ±1/1{3 ). ~~~A ~o.«, ~' dl ~~ - - 2(1-2~-Z)-ttl~i. : 2. < 0 ~ A yY\&61)(~. 3

>

c1-

I - .2A., X. -A. 2. 0 I - 2.A. a"J - A.2. = o f ert::: l - 2A.1~ o ~' = ?G'2.+ 'd'Z..-t =I

f; =

rp~ ~

31z.

'

~ 'X.=~ =o D.Md ~=±I

=

= 'X.+~ = 0

22. (a.) T~ ~

21 ~r

=

f~ =

~2..

~=~, 'X..'l..+2-llJ't.=I,

X,=11-2'dt.. f'=t<-~)C1-2~?.r•1i.+2=0, 3'd't.=1, 'd=:t11.f3

41-2':3?. c,:;.~~> '2. . ,,_2A},,):r2. 5• = x+"3+c -:-'A 1 lx.,_+~'L+??·) -A2.l~+~)

.Ao

1

(1}(:2: )-A. 1 (X.'Z..+>j'Z..+"i,,)-A.i..('d-r_)

f"

(cl)

A,= ''2.

~' x=-2/"3, "J=r= vrs ~ x=-+ 2./(-3, "a =e:: -1/"3

2

A:Jo c

=e·

=

AC =___.a.;;._

c to BAO=

~ = __..b,___ "'"z. rv~ ~(3

k

l\Ji

l\Jj Ct'l ct

J)

1{

'T' = Ni~cf + ~er..p = ~

b

B : J..

A.6-c~~ ~~~ 1 ~10~1\ :oc+cE ~ ~:

~ :: 'DC+ CE =a.~~+ b~~ ::: ~-

p~d >

'

d.1' = ~ ~ d.~ ;.ll. ~~ A~ "'i Cr.:>'Z.
'*~

(J

s~ ~ol) d.f

M.t

:: JL

~'-ct

/\J'"z. ~'l

clct -+ ...!!__ ~~ q:,1..~

(""

p\

=0

=0

~ ~~ 1.t.h ~, ~ ~ i{d

Section 13.8 248

(b)

Section 13.8

Jt -x.,t bMx..JJt. = cl St ~t..-x.. ~x,tLt. =Jt (J...x.)et.J....:x.A
I. Cb) cl

d.t 3 _l

Cc) a.

S'2- e~ "'4-3 = - OllcX.) e~2. = - e ct'Z. .J

i.

~cl !

Ce) ~

~

f 2X .k l.u.t.+ 't.-a.)tL.t. = J2.~ 2;.x

n

+ (2.) k('l"-~'X-t.)- Cl) ..tM(x,"L+x,z.) ~ M:+x. =S~ \'X. z.cllt+a.~x.+k~

x.

z. cL..l

~

cl,.2.

M+X.

s ..tn(.u.'l.+ x,'2-) cL.t = 5x. (~ -(u..'-+x})-z. q~ ) ci.u + (2.) 2.~ ~

~

4"1.. x

: sx.

t.

M.~+'X..~

4x.z.+~z.

2.~ 2.(Mz.-~z.) clu.
NOTE:

CC)

+ .2,.

s~

dt/(t3+n, -fc~J= <-~~ )(1+eo3 x)' - C-1)(1-x.3 f', _ 1 11 3 -Xf l'X-) =- ~'X. (It Cf'::> 'X,) -~'X. (-1)(3)(-~'X.. )cr1°X. (I+ er}~) 2. + (-1)(-3'X-'2. x1-x.3 f2. fto)=J 1 dt/C:.t 3+1), f'(o)=-1, f"to)=-v2 , o Ao fM (s~ cl:t/(:t3+1)) + ?C. - lf x.... + ···

2.. Cb) flx)

µ)

- (I)~ + ~ x'-+'X,.,_ x.

fc~)=

=f 00~

=

1fu ~

~ ~(l/ct:'3+1),t=O .. l)j ~ So' J..t/<±3+ t) ~2. + ,a,n.

=

2.AAM'X-

f0

kct\1)ott J f 'c~)= (2.Cd)X.) ~(4~'X.+ l),

AO fCo)=O,f'fo)=

5-''C't..)= -2~~ k(4~X,+I)

o, f ''fo): 2,

AO

fC~)=O+O~+

.

+ 2.C4:)X

;! ~i.+···

8er.>X.~~+I

4~-z.x.+1

Section 13.8 249

5.

so

oO

=

J(o..)

(Cl)

4'.

~ JJi_ - _ir_ _ t+~:s - 3~(~n)

=-1L,_

J'ctt> = SO() ~ct~-x. ~ o

J''(°') =

I+ x,.3

~A. (kx_)"-~ =

So0

0 · l+ ~ 3

J0

cl)

NO'W" Act o..=o: (b) ' · (tL)

-r

'

_'IT?.

~

= 811oi;;3

~

a.=I

ff [-~(~n)) ~(Q.±!n)

l

'

J'~X. H:.t>ctt -t C~fl~} f(-x,) v

(~"'J (X.Aj )''

=

(rx-t) fCt)t!X , fCt) Gl:t + (l)(,,~·~).f(~) )

3

0 ./

.

~'= t.f:··· 11

d

~Co.)= ~'(~)

=o ../

o"

~'<1t> =t s1··· = 14 'Ctt) = f''' ... = o v

s:= SA.. . =... = v

-~

o. o

0 .../

ff~) ./

s:··· =' ..;

d(~) = e-t- + So'X- t'l.~('X.-t)ctt. ) "a'(x.):

I

1,

' cs1lf )

o 1+ x.

,

s: )' = f(l.x, =

'X"(1('X)

~!(Q.+lrr)

l °" ~~; M. =_1r2.(-l/2.) = ~z.

(D.):.

=

+ CD(~rr)<-2)J C
3

-t f: (~-t )-s fCt)cd: , 'd ts:··· = t J:?C. 3(~-.t)?..f(t)cl:t + = t.a.s: 2,.('X,-t)flt) JJ:. + fc~·~(~>, "Jd,,,, ('X,)

(C)

(a..)

~

'd("-} ;: ~'l?C)=

rl''
'Z..

(Q,...x.) a+x,3

FJ\4W\. ~~,A

~<~rr)

~'2..(o._;'n)

'

~lo)= e 0 + 0 e'X. + S" :tz.~ l~-t)d:t + ~2.~C'X.-~), ~ 1 (0) =e +o:: I v 0

~"(,:)= e~ ... s:x'l.~(l'(.-t)J.t + X.~X.)+2.X.

="alX) + 2X. v

.

7.

JC11.) = s~ 'X.a. ~ = Q.~I I~ J'Ca.) = s~ X.a. ~-x..lJt = -(£1.~l)'L ~~ ct= 0.7 ~ ] 0 x,0~ k'X,~ -1/2.i' :: -0.34'102.

8.

Jca.> =S0 1~ k

=

ct I

I

> J'Ctt)

\k~!. ikt J.
II. (b)

3

s~ \:~ ~

M(?lJt) Cl

110

l

= S0

n

dL~

?<;

;7Jt clJll = ti

1

J.<11.) = .x.v.
Jcoi=o= t,., + C ~ C.=o. Ht...q, Jc11.)= L..
=Jl3) = ~4

= R~-ct)+F
«.+c.:t

+..Ls. 2C ~ct

G
M(~,O)= Ft~)-+ ic. ~

r: ,._

=FC*Xw)

~

Section 13.8 250

=

~" [F'cx-ct)+ F'<~+ct>]h. +

tc, lGC-K..+ct)-Gl«.-ct.>)

NOTE: F'{4"d) ~ cl.F/cl~. Fdl. f4..) F'c"-ct) = clF<~ct)/O.c~c.t) ~XJC. =[F''r~-ct)+F''C~+ct)]/.2. + [G<%+ct)-<='<~-ct1]/2c ';J::t = [-cF'112. + c [ Gcx.+ct)+ Gcx.-ct >]/2c) ':1 ('X,o)::GC'I(.)..; ~tt =- [-c<-c>F''c'X.-c.t)+ clc) F''(-x.-+ct)]/2. + (ci.G'c~+ct)-cz.G'c~-ct)]hc.

~~ (C)

.U(X.,d)

A..l-x. =

~~

MJ(

'dtt =c-i.'d"-~· v

J f(J) cl~ -- .&JI fl~) ~~ 1t •I (!-X)'t+~i. .1f' •l (X-~)£+~-i.

=~

1

~s l (-ff?.)(~-!) fff) tA~ Tf

-I ((X-~)z.+ "jz. )2·

') )

U

:

- 2MJ

1

-W -I

XX

..if!.l. tA.1: + d-J 1 4(2.)(X-3 f-fC!) tl2 (

JZ.

:>

ft

•I

[

)3

::>

.u"" =.Ls' _fill d.~ + ~1· c-n2~ f~)t!2 1T •I

d

t: J

1t

-I

L J2.

'

.u'i)'l=ts~fl~>a~ -~J'~d.~ -~2.r c-a)(23~)fa~ d

-· [

]

. it' -1 [

]

rl' ...

[)

1 crd =- ~r Jfff c1.~ + ~s' ~:f<~).:1.~ = o Vi(JL lF~

Ao Mn+M ..

if k f I

12..(4) MCX,"a) =

0

[C'X-~)'L+(~-~)-a.] :ftJ,vi_) ~~IA~

J

Section 14.2 251 CHAPTER 14 Section 14.2 I. .Jd.X~

~ "Jo -!:!- i:-d, I:[, /lo .J: C
¥

2... ~x~=Q~~ ~~/dt~~Q ~ ~~~~~~~­ rr~ ~ ~ ~ ~ ~ .u..·Ar=o ~ -rk ~ ~- W.t.~ ~ LJ.. ~/cit M" =0 . - "'"

3.

"'

=,..,_ --11£-U'Z. = Ilg.II?.+

C·C.

(a.-b)·(a.-b)

~

=-a..•a.+ JI~& ........b·b_ 2.llt11lllb ,.,, ,..,

11 ~1!'2..-2.

4.

ll~U llhH CD e

o't)

£X(£-s)~fX~J ~ £X£=£X,S-) Cb~Q'..~:ca.~('fr-e)~ ~ ~ ~~1 th.Lr~,~ h~~=a.~crr-~)

=£t. ~l ,

5.

=

~, c'2. a_Z-+l?-2a.b~-9.

.u.x~= 11.ttlllll\J"'U~e-e ""-'"""""" ...... "'-"

~ Jl.U.Xl\T"ll?.. ~"""'"

,oo _/J,,_ ~ot

=

=L

~(3>

l\.A..lll'Z.HJ'j\l'Z.~'2.-e ,,.,,,, ~

:: ll .u. lfz.. I\ N' /( -i.. (

I -~~e)

= u~t\z.ll ~ur.- (~·(\j)'Z.. A.,

s. No.

f.o'"I.

Mt. O.o.

""-

,.,.,

1

~f, ~- ¥,~~ ~ ,.,;...~ ~ ~

......,

t-'""F- ~

~ oJ: 'f4.~J ~ (~~l\i)X~= f'x~=Q:

~ ~X(J!'X_bf") ~ ~ ('1.Mtl ~~~:-!,).

~- Cot~)x!f

= llagnn~u~e g (e~tott.vt,~~)

-

= a. n.u,u U1'rll ~e e ,..., -

L.u

Section 14.3 252 ~ ct>O, ~

.

I\

(d~)x~= lla~llll~ll~l~-e)C-~)

-

- l\rj\- ~e e = Ol Jlu lllt

= IOll ll.u.H llN"ll ~e c-e)

*

ei
l5.f~, 1 o<::O~"(Sb) ..<. ~ Q-=Q. 'l'-lw:i., (cl~)x~=«(~xffJ

~-~~~M4'.

10.

Jt~~~ to

(4)4 n=2..

w,...,_

Section 14.3

/\

~

~ ~

Ne4,~ it~ fdl n:--k.rr~

i -

II -I 3 I I -J

A

= - 2-t -4- 14 A + 12.'£-.e. /\

I\

"'

A:

-

II c~+3~)x~ll =~C-2)-z._,.C14)i.+(•2.)-z. = ~3ll4 = 248'1. ~- ~

= (2)°3)+HXo)+(-3X2.)::

m s~~J ~'E" ~~ tyx~ ~ ..lO

0

J

l~·~r

=101 = o.

°t? ~~~Ci J:~~ ~),~ { AM. ~ ~ ~ ~. Lit .M.6. 4U..: J..

~·(~x~)=O ~ ~

Section 14.3 253

Section 14.4 256

Section 14.4 1. Cb)

rvxt,.J= ..... -

·~~t

Ii

tJl,t j f, -31 -~1+2~+2:k, .u.x.{/\T'X14")= -f ol ::-2t-2~-.2.l d ""' "" -JI 2 z. 4 1 ~ A. '9t, ,..I ) '"' 0 1 = -3.l-3~ + 3{ (.U.~l\f)~Mr = 1-3-~ "',..~I 3 =O..t-3A -st t. LAX.(~x~. 2. I 3 d ' ....._ ,._ ,..._ O I -I cJ "" ""' ,... 2 0

J::

A.

MXl\r: ~

2..

(~)

.....

A

A

'

3. Cb)

4.1.k. (5). I Cb) V~:: 1 0

0 0

I 0 I I l

(C)

V~=

I = 11 l =.1

11j; ~11 =1-"I =" 0

I I

A.

A.

-""'

~8xf3~.L:to~,,60 8)(~·~=0 CAxB+ BxC.-+ .....cxA)· o-o+o- ~)(f·~-+ ~xB·~ -o ,.._ "" ... ..... ... CB-A)= ..... "' = - B· ex A+ c x. A· B = o J


- - - - __.......

Section 14.4 257

AxB·C-o +o --@x£.·8 +o-o

-- = AxB-:c+cxB·A

= AxB·C+ C~B~A-= AxB·C- C·AxB = O ./ ""-

dt

....

,,,..

""""

"""'

""-'

4"-

""'-

~

Cb)

11.

)

................

-

.A.l .

,.

,..

/

- - = B·@X(~xl') F(7) = B· cc§·~)~ - c§· ~r~ J ~ c') = (A·C)(B·:D)-Ul·D)(B
~.(a.) (~x~)·(G><:Q)

-....

~X.(.U.Xf\f)

v

1

~~'~ -il•!!X'!=O. t'k~.1~,k~1d~~4 ~>~~ M\ Q, - ~ ~tf..t ~ Ar.. LD !1J ik.-.. '3.rd. ~ E,tf.!{ AAe. ...U ~ ~ ~~ A4 M~ ~~ t;f;~ ~ (i ~~ck>~~ 4.

~~!1d ~.k ~ ...,,J.~ L]))} ~ ~·!!'X.1.f=O ~ :tU ~ b ~ i-k M",1£ ~ AO ~'~ A,g' ~ ,k LD. ~, ~~ ~>'>!,~ A1U. LJ), ~~Mt<.~~~ ..u. = o1 rv+ n.w-. 'f~~·~x~= o1.W~+f~~~=o. 6.n,·1-

- - '"""

~= °"';6+ f~

~ ~·!'.J"x..!_.r =-~·~><~ =-~x~·~ = - ot.~y.!'!.y..-~~~¥= O. 6.n, 1 ~= ol~-t ~~

Section 14.5 258 ~ M·~X>t!"=¥Y..f!"•(ol~+e-~)=0-+0=0> Ao~~~ u f\T: Mr .. .:....~~ --tW .u.. /VX.W- =0. ./ -'~'-

~·--r~-

B

12.

- - '1'-k f>~ A,B,c;D L

1-

°4 ,.__a~ i °f{. ~ Ak:::;. ~~. a~,N:? ~, A.e., 1tS ""'-~ ~ ~ L"t>.1'~, A B, c;:D k a.~ 1 ~ ~ { 8~· ~x~~ =O. / " _..c

~

AM

1

( b)

M- "-

8=(:Z,1 -1), ~= (1,3,0), ~ =(5JO,"), 'P= (P,0,4), 1

-r

-i

t.~.e.J ~1'

Ao

2 1I

8f}·8c;x~ = 1 :~ ~ = H)C5)-c2)(3S)+CIX-S) =-ao:;o MJ

Cc)

J ....

B= ('l,O,O), ~ =(o, 1,0), ~ =(1,2.,-4), :p =(0,0,1), MS

A)B,c,D M~.

~~-8~xffp~l1~-f I= r-4)(1)-(1)(-1~)+0 = 11-toM>J\,8,c,D4 ~­

Ce)

8= (1,0,1)> ~=(2,1,3), ~=(1,-1,0), l?=(3,-J,2), /:lCJ

~~· 8£X N~ = JB-t I= 13.

L.ai:

=0

(I )(-2.)-(1){2.)+ (2.)(2.)

Ml

A,B,C,])

~ ~-

~ = (1,0,0), ';!" = (J,1,0), '1!" = Cl,1, I),~- rp~ ~X(~)(~)

=l~·~)r:f"- (~·~) ~

CD

-

= ru-w = Co,0,-1) ~

~ (~x~)x~=-~x(~><~)= -(~·~)~+(~·~)!)[ ® = -2~+~:: (-1,l,O) =1= ~X(~X~).

'l'o IW\~ ""f ~ ..ux~ ~ ~ ~, ~ -fN'M.
tWMr(

~10~

(~![- (~·~)~ =-(~·~!)~ + (~~~

o'L,

.

(~· !!)~ = (~·!!)~.

®

rp~) ~ ~ ~ ~e.ut. ~ ~- Lit .y.=

~ ~ AO &..o

.

AA,{ CAM.

~

(1,0,0)

c~ f".-0n.uJk1

+k ~J:t. ~ ~ ~ ~ ~14.

;to ~ ~). rp~ ~= (o/ 1 D1 0), ~: (olfi,o,o) = (o1p,o,o))

(P','if, b) 1 ~@ ~

~~~-fdl ~ o1,~ 1 'il,S. hfl~J ~

=

(1,0,0),

'::! = (S, ~,-3),

~: {4,o,o).

Section 14.5 l. (h) ~: (C)

,1- e~ ~ ) ~I: -eT, AJ.. :: 't~ t_-4~ + 3~21:: k ""' d -J II U:' II = ~4 + 14'f-c.t:>'2.2.'C

r) AJ.. -

1

J.J..

11

= -e'C

1 ) lli-t-

=zt1-'1~2't t

,

JI = e'C 11 .U. = 2~ -12C(.)ZC t 11

-

-i: fcr.>~ ~2-c Ce> ~== e-t(C1":>2tt+~2'C1), ~'= e [C-c-2s)i+(-s+2.c)J],

Section 14.5 259

et et

2.. (C)

~,, =e"C [(3s-c)t-(s+3c)a] ) ll~'' 11 = 4(ss-c)'"+ (S+lC )t. = 4C)Si.-~ +ct.+ s<+~Ci' ,C...,. = ,/70 e-'t W~ JO ~d (?e): (~·t:fX':f") : ~'·~X';f" + '!·~'X~ + zt·~X~' J -f
1

""' ""'

=I

"""

-'t 0

0

2.C 0

s

I=

~ ( ~·':f X':!)' = c ('t 3-,'tC) + S(3t'Z.-,c+,ts) 21: -I ~'·~X!:!: 2C 0

-3 -'t

0

s

=-5 (-2'[.'~+ 'C)

0

~·~''Xti=-1-;;-;-:~1= -s(-t-z.-"st) Th..wiAuJm.~ 3S1:t.-,sc+'s'Z.t+ct~,c~ o

o -3'(:

"(~

-!

2C

0

0

c

I

¥•t:JX~:

:C('C3-,tc.)+

S

5(3't2-,C+' St)

3

./

-t = -c.(-'C +'ct) 0

~' ·..x N, ~:to~ £1e),,;,.. ~: ( ~·~><~)' =

3.(Q)

.U/t) .Ui'L)

~ft) ~l'tl

~Ct)

.w; (t.)

.u 3ltJ

..

Af3C't)

J

A.U3(t)

~ AAA-l (1"·3) ~ E)(~ '" AM. S..t.c. I O.tf- ) ~ AAA(. ( ' ) ~ S'..u:.. 14.Lf-.

Wr.vvJ: to~ (7C): (~·.y)' = ~· ~ +~·r:[' ~ =31:~ +t~.«.) ~ =t-4t~.

=(3-C)' =3 ,

u •1\T+..U. ·rr' = 3+0

= w~~ il~ (Jell: (~x~)'= ~'x~+~">'.~'
....

.....

(b)

f-c

1

c~x~)':: *-c

n I

1

6 t"'I = f't(tttS't+ 't't1-12t~1) = 20t.4 1+4-c 1-24-rt

-4t ~ 0 A

¥ X~-t-~X~ = ~ 1

1

/':\

-

~t-'O

=~

;r. b ,..

b fc3

+

I -lft 0

4. (Q.) (f.t.l)'= ~

3 ../

A

0 -4

t

'trr:

3

A

0

4

=20t4 t +4t.3f-2.41::t

f{t+~)~lt+~)- f(t)~ft) ~

fet-r.ot)~f'CJ- f(t)~C~) + ~ -.ftr+.1t)~(t+.6t)- fft-rllt1¥lt)

At~o

~'t

t;c

At~o

= (~ :f<'t+41:)-f~)),U['C} + ( k :f<1:+A't)xk ~tc+~t.)-~(~)) ~...:,O ~t. .6't~O .d't,_,0 .1t

=.j. C"t) ~l'C) + fl-C)H'C-c > v' 1

5. (tl..) (~·~~~)' = ¥'• t!°Y.~ + ~·(J!X':[)' tt11 (7c) = ~1 • f!'X>;t! +A!:('f'X':f+~X.~ 1 ) fWl {17&) =

1

1 J..l •Af'>tW"+ J..J..•f\T XW,,,,.,,,, """"' """"""' ,.,,,,., ...... ~

A

l't. A.+Lft. j'-12.t~+lft ~-l2.t.f 1

4

1 + AA.•f'S'XW,..,,,, .... ,..,,_

.../

V

Section 14.6 260 (b)

[~X(~Y.~)]'

= ~Ix(~)(~) .... ~)( {~X~)' F (7d) = ~ >( {~XI!!)+ ~><(~'Xfd"+ f!X~') 't"-(7cl) 1

1 = .u.'x{~+W-) + ..U.Y.(N" ><.Lv)+ ux(N"xw-') ...... ,,,,,,_ ,,,_ ,.., ,,,,.,,. ,,..., ,,,,,__

""""'-

"· :'(. ~(f<-cl) =

1t.[

f

~

l)J

+ A.lz Cfl't + .u 3 ( f('C. \) t] ,,. .U/f(t.}) ~ + ~ il2.(-f('C))~-+ *.tJ.if('t))Ai_ A 1::: .U.~(f) r!f t + .U~l-f)!!f~ + .U3(f)~ ~ at. c:l't,
.l.l.1 ( ('t \) 1

= Ii

A

./:.

CA\.

"'f c*'t

7. (b)

(JJ..Xf'IJ") ...... -...

11

= (.u'xAr+.U..Xf\T')' =(U1 XN")'+ {A.O(.~')' ,._ ...... --... ........ 1 JJ..'' x rv + u~ x l\f 1 + A.A! x f\i + ...u x ~" ........ ,.,_ -.,..._ ......

=

=

9.(4)

ll~et)ll = (~·~)•/z.

""'-

~

11

- -

J.J.. x N'"+ ......

A-a

~

"'-

""--

2u'x l\f 1+ .u..x. l\J" .....

......

Jl~(t.)l\ 1 = f'(~ 1 ·~+~·~1 )/(~·~)Y?. - J. 2.u.'·..u. _ u'·u -2. .... ~u·.u n.u II

- "'"

(h) ~ \l~C1:)\l

Os

...

-

= ~~ :FO ~ (S.I) ~ 0=~·~' ~to~ dt .cl:k Q.

Ao

~1 Ai;)~

Section 14.6 I. (b) Jt= ,/2.z. + C-l)T..= (C)

2..Cb)

te)

n=4 (-3)t..+ 2.t.

r=

3)

f5, ~· ~: 2'S7°, AO" -6= 3'0-2'.57 =333.lf-3°: 5.81~.no..d., 'l= 0 =ffl, ~·~ = 33.,, 0,~ &= 180-33.,,= 14'.31° =2..554~, i!= I

cp=-,o()= 1r/2.~) -6= ~o·:: 1T/2 ~ ~1 13~G,z. ~

c

~1

1

=~ ,45 .b = 8.48° AJ 4'= ~0-8.46 =81.52°=1.423.n.Ad.

i: 2<-5"1° ~ e = 3b0-2".5'7 = 333.li3°= S. BJ'}~ ~

3 to.M.' ~ 2.i._.. 4'-

(.i)

~

f=4f>t..+<-3Y-+1i. = 44'

=.33. 85° A
~I~: 2.'.57°

fto

cp = ~o + 33.85" = 12.3.85° =2.1"2 -'lA.d. 0

-&: 3b0-2,.5'1: 333.lf3°

=5:81".no.d.

cc)

,. . /\ 1 R ,,.. = .x.t.+'(1j + ~1t-

= c.t-~.t +t-z~ -3t:ft. ~= ~ =(2t-l)t +2t~ -3{ = e.n.-~e ee )+2t {~e e.11.+ ~eee)- :?e~ '2.

,_

A

~

(2t-1)(Cd:>&

~ e&>r!J-='X./Jt.

= (fl.-t)/~(fz.-t)?..+(t-z.yz·;:;: (t~t)/~2t""-2t3 +tz. = (t-l)/42ti;.2t+I ~ ~f7:M/J7..:. t 2/ P =t/-/2~~-2t+I , AC1 d rv = 'ft -3t + 1 :t ,,.. ,,.. 3 ~

A

""'

f2±-z.-2t+1

~=

eJl. -

b..ti._2t+1

~e - ~~

2t+2j = 2(Cr.>9 e.n.-~ee)+2.(.AMMee)1.+coeea)

= 4t- a. J2t?..-2J:+l

~ '-Jl -

2.

{2ti.-2:t+ l

e"" &

Section 14.6 262

ccLJ Tk

CO\.(..

.t4, ~ ~ •

d

J.>rl..

A

~ A

ALt-

1'4.

A

T~C23)~ t;[= oe.n.+(3Xl)ee+'e~

~ (2'f) ~ a.= (o- 3Ct)i.) A.. -~ , . ., Jl\ (e) g=.rl+'dj+~l = 1:4ra+2~

r-r=R=lfi "" /\S"

~

~~

=

,,t e.n.

~1,tz.+1

= 1/~

'"tz. + r

e

s, .e- =t, ~ ="r ~

ee

=-3~

~ ~-e- = 'd/JL =Lit/,/ 1~tz.+ 1 ) MJ

v1"t?..+1 -e

~= ~=41=Q~} ~'l ~~""'

A

= 3ee+'ec e..11. + (C3Yo)+2loXI)) +oei:

=4(~9e.A.+~ee)

er.>&= x/J1. + 4

.Jt =-

Section 14.6 263

..

,~,e>

c A

aer =~ ~

A

~

eff(34>+~ 1 e)-eeCB4>,B)

t4>~0

~

=£4>.., ~0 ~e"' = @~ 4Af'

't'

Section 14.6 264 Ao


Ms<~~ "de{'/?J0

~

= ee) ~NA. ckJ (~ ~~=~1T/2. =l ) . ./

V~1 ABC~:

A

A

.

~ €q,Jq,~ ~l
A

c

~it=

!io §1,icp+~- e.p\g, = h

~~I~uq, /

~~~~

(~e£)

-ect>

=

NOTE: ~ 1 ~ ,.,--t,.~.. ~--it{~-~~J .wt~ clo ~ ~

'tQ,

~:

ee= erx e