PRACTICA 1 I.SOLUCIONES DE ECUACIONES DIFERENCIALES 1. Demostrar por sustitución directa en la ecuación diferencial, comprobando las constantes arbitrarias, que cada primitiva da lugar a la correspondiente ecuación diferencial. SOLUCION:
a) y =
senx +
–
x es solución de (1 x ctgx)
Solución:
- x + + y = 0
x cosx … 1 x … 2 … 3 − − − − − … 1 − … 2 y=
senx +
=
cosx +
=-
(1 + x ctgx) -x = - x (
y=
x
senx
= (1 + x ctgx) (-
cosx +
) = -x
senx) = -
senx +
cosx -
senx +
Luego sumamos (1), (2) y (3)
(1 + x ctgx)
-x +y=-
senx +
x cosx -
(1 + x ctgx)
b) y =
+
x
+
x cosx +
x+
senx +
x
-x +y=0
+2
es solución de
-
-
+y=8
Solución:
y=
-
+
x
x
-
=
+
+
=
+
+
+
=
+
+
+
+ 4x
+2
=-
-
-
-
+
+2
+ 4x
x
+
+
x
-
+2
+4
x
-
-4
+ 4x +4
- 4x
+ 4x
+4
- 4x
-2
+2
+ 4x
+4
+ 4x
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− − … 3 … 4 − − − − - =-
-
y=
+
-
x
x
-
+
- 4x
-2
+2
Luego sumamos (1), (1), (2), (3) y (4) -
-
+y=
+
+
+ 4x
+2
+
+
-
-
x
-
+4
+4
+ 4x
+
4
+ 4
- 4x
-2
-
-
-
x
-
+
-
x
- 4x
-
-4
-2
+
- 4x +
x
+2
-
-
+y=8
2. Demostrar que y = 2x + C
es la solución de la ecuación diferencial, y
- y = 2 – 2 – 2x 2x
Hallar la solución particular para x = 0, y = 3 (esto es la ecuación de la curva integral que pasa por
(0,3)) SOLUCION:
y = 2x + C
=2+C
… (1)
-y = - 2x - C
… (2)
Luego sumamos (1) y (2) -y=2+C
- 2x - C
- y = 2 – 2 – 2x 2x
(x , y) = (0, 3)
La ecuación de la curva integral es: y = 2x + 3 3. Demostrar Demostrar que y =
⟶
3 = 2(0) + C
+
+ x es solución de
C=3
-3
+ 2y = 2x - 3 y hallar
La ecuación de la curva integral que pase por os puntos (0,0) y (1, 0)
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2 1 4 … 11 3 3 6 3 … 2 2 2 2 2 … 33 ⟶ ⟶ − − − − ⟶ ,, 1, 2 2 1 ⟶ 2 ó : 2 Luego sumamos (1), (2) y (3) -3
+ 2y =
+4
-3
-3
(x, y ) = (0, 0)
0=
0=
+
+2
+ 2x
+0
= -
0=
e-
-3+2
+ 2y = 2x – 2x – 3 3
+
(x, y) = (1, 0)
0=
-6
+
+ 1
+ 1
=
e(e – e(e – 1) 1) = 1
=
La ecuación de la curva integral es: Y=
+ x
4. La primitiva de la ecuación diferencial diferencial x curva integral que pasa por el punto (1, 2)
SOLUCION:
= = y es y = Cx. Cx. Hallar Hallar la ecuación de la
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SOLUCION:
=-
senx +
cosx
=-
cosx -
senx
=
cosx +
… (1)
senx
… (2)
Luego sumamos (1) y (2) +y=-
cosx -
senx +
cosx +
senx
+y=0
= A cos(x+ B)
= - A sen(x+ B)
= - A cos(x + B)
… (3)
= A cos(x+ B)
… (4)
Luego sumamos sumamos (3) y (4) +
= - A cos(x+ B) + A cos(x+ B) +
=0
Ahora demostraremos que y =
cosx +
senx ,
y = A cos(x+ B) son, en
realidad, una sola.
y = A cos(x+ B) y = A cosx cosB – cosB – A A senx senB Como A cosB y A senB son constantes, constantes, pueden pueden asumir el valor de
y
= A cosB
y=
cosx +
6. Demostrar que ln(
) + ln
= - A senB
senx = A cos(x+ B) = A + x se puede escribir así
= B
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.
ln(
)+ ln
ln ln
= = A + x
)=A+x
= = A + x
+ ⟶ 1 √1 =
.
Como
=
es una constante
Reemplazamos en
.
B
= B
=
=
7. Demostrar que arc senx – senx – arc arc seny = A se puede escribir así x
– y – y
= B
SOLUCION:
arc senx – senx – arc arc seny = A Derivamos:
√ −− −− √ −−−−−√ − −−√ −− 1 √1 = 0
= 0
= 0
Integramos:
∫ 1 ∫ √1 ∫ 0 1 √1 -
x
- y
=
= 0
8. Demostrar que ln(1 + y) + ln(1 + x) = A se puede escribir como xy + x + y = C
SOLUCION:
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ln(1 + x + y + xy) = 0
Como
⟶
= 1 + x + y + xy – 1 – 1 = x + y + xy
– 1 – 1 es constante, constante, entonces entonces puede puede tomar el valor
– 1 = C
x + y + xy = C
9. Demostrar Demostrar que senhy + coshy = Cx se puede escribir como y =ln(x) + A SOLUCION:
senhy + coshy = Cx
− +
= Cx
= Cx
ln Cx = y
ln C + ln x = y Como lnC es constante entonces le damos el valor de A= ln C y = ln(x) + A II.
de las ecuaciones diferenciales
1. Se define una curva curva por la condición que que cada uno de sus sus puntos (x, y) y) su pendiente es igual al doble de la suma de las coordenadas del punto. Exprese SOLUCION: La pendiente es
= 2(x + y) = 2x + 2y
m=
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− − −−−−− − − =
=
=
2. Cien gramos de azúcar de caña que están en agua se convierten en dextrosa a una velocidad que es proporcional a la cantidad que aún no se ha convertido. Hallase la ecuación diferencial que exprese ex prese la velocidad de conversión después de “t” minutos. SOLUCION: Sea “q” la cantidad de gramos convertidos en “t” minutos, el número de gramos aun no convertidos será “(100 – q)” y la velocidad de conversión vendrá dada por q), donde k es la constante constante de proporcionalidad. proporcionalidad. 3. Obtener la ecuación diferencial diferencial asociada con con la primitiva y = A SOLUCION:
. = A
= k (100 – (100 –
+ Bx + C
+ Bx + C
= 2 Ax + B = 2ª =0
4. Obtener la ecuación diferencial asociada con la primitiva
Siendo A y B constantes arbitrarias arbitrarias y “a” es una constante fija
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5. Obténgase la ecuación diferencial asociada con la primitiva SOLUCION:
− − − − − − − − − − − − − − − = A
+ B
+C
… (1)
Multiplicamos y
=A
con la ecuación (1)
+B+C
Derivamos respecto a “x “ - y
= A
- C
Multiplicamos - y
… (2)
con la ecuación (2)
= A – A – C C
Derivamos respecto a “x” - 2
-
Multiplicamos - 2
-
+ 2 y
= 2C
… (3)
con la ecuación (3)
+ + 2y = 2C
Derivamos - 3
+ + 2
= = 0
6. Obtener la ecuación diferencial diferencial asociada con con la primitiva y =
SOLUCION:
2 1 21 11
+
+
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=C
+
= 2Cx = 2C =0
8. Halle la ecuación ecuación diferencial de de la familia de circunferencia de radio fijo “r” cuyos centros están en el eje x. SOLUCION:
+
=
x – a – a =
1 – 0 – 0 =
− −
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SOLUCION:
= 4p(x + a)
Derivamos:
+ ++−− 22 22 = 4p
= 0
= = 0
= =
PRACTICA 2 I.Separación de variables.
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SOLUCION:
1 11 1 11 0 1 1 0 11 1 1 11 1 1 1 1 1 1 2 | 1| 2 | 1| 4 40 4 0 2 2 0 2 || 14 4 34 4 3 4 3.
SOLUCION:
4.
SOLUCION:
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1 1 0 0 1 1 0 1 1 0 11 1 11 1 1 1 1 1 2 | | 1| 2 | | 1| 2 11 1 √ 1 0 1 1 0 1 √ √ 1 6.
SOLUCION:
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1 1 2 2 23 |2| √ 4 √ 4 1 0 0 1 1 1 0 |10 1 | || 8.
SOLUCION:
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1 1 1 0 || || | |||| ||| | | | 1 ∧ 0 1 10.
SOLUCION:
II.Reducción de variable separable
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12 2 2 3 | 2| 32 2|2 2 | 3 1 1 11⇒ 2.
SOLUCION:
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0 0 0 1 4.
SOLUCION:
1 1 − −+ − ⇒ 1 −+ −+ −+ −+ 1 ⇒ 1
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3 2 3 30 10 0 1 1 | | 1| 1 6.
SOLUCION:
7.
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→ 14 − 4 ∫ − − − − 2 0 2 − 2. − − 2− … ℎ − → 3− 13 − : 13 2 63 ó : −∫ ∫ 2.
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3− ó: 3 → 6 316 −− ∫ 121 − − : − − 12 12 − 13 .. − 13 12. − − 13 − 13 12 1 2 … ℎ: − → 3− 13 − ó : −∫ ∫ 3.
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21− : 21− → 2 − → 21 − 2− 21 21− 2− 212 21 − :− 21 − . − . − − − … ℎ: − → − − : 2 4.
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ó : −∫ ∫ −∫− ∫− − − : − :− 110 0 1 0 1l n x − 1 .. − 1l1lnxx. − − − 1l1lnxx … ℎ: − → 2− 12 − : 1 1 5.
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2 21 − ó : ∫ [∫ ∫ ] − ∫ ∫ ∫ 21 − 21− − 2 : → 1 → 13 − 2 − − 23 49 − − − : − − − − 22 2 2 6.
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− .− . − − 1 − 12 … ℎ: − → 2− 12 − : 12 1 12 2 1 −∫[∫ ∫ ] − ∫ ∫ ∫ − − : De la ecuación general:
+C)
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… ℎ: → 2 12 : 12 22 − ó : ∫ [∫ ∫ ] − ∫−[∫ ∫− 2] 2. ó: → 2 ∫
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: 1 2 2 2 … ℎ: → 2 12 : 12 2 2 − ó : ∫ [∫ ∫ ] 8.
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− 9 9 ∫
: −− 99 22 22 919 23 2 618− : 6 18 0 2 2 − − − − − 12 … 9.
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− 1 −∫−[∫ ∫− ] − ∫ ∫ − − − :− 3 313 13 3 3 3 3 3 3 − − 31 3 .− . − − 1 − … De la ecuación general:
10.
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− − ó : ∫ [∫ ∫ ] − − ∫ ∫ − 3 −+ ∫ −+ − 3 − 3
ó: → − 31 − 31
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23 .. 3 −. 32 3 …
ℎ: → 3 → : − − ó : ∫ [∫ ∫ ] − ∫−− ∫−− 3 − 1 3 ∫
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− 21 .. . − − 21 − 1 … − → 2− → − − − 1 1 2 −∫[∫ ∫ ] − ∫ ∫ 2 − 2 Luego hacemos:
Reemplazando en ( ):
De la ecuación general:
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ℎ: − − → 2− → − : − ó : ∫ [∫ ∫ ] − ∫ ∫
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− + − 1 1 1 12 1
− ó : ∫ [∫ ∫ ] − ∫+− ∫+− 12 1 112 1 ó: 1 → 112 1 116 1
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SOLUCION:
− 0 … 1 : − 0 … 2 12 2 0 0 2 0 : , . − , 2 , 2.
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: 2 0 … 3 02 0 ≠ ,, . , 0 … 11 : 4.
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; ; 1
Resolviendo se obtiene:
′′ ′′ 1 1 tan− 1 1 1 1 12 ln1 tan− 2 ln1 ;
Luego:
Luego la solución general de la ecuación diferencial diferencial es:
14.
SOLUCION:
El polinomio característico característico de la ecuación deferencial es:
1 0
De donde: r = ± i
Con raíces imaginarias, las soluciones son: cos(x) y sen(x) , luego la solución complementaria:
= cos sin
Determinamos la solución particular. Para ello determinamos el wronskiano:
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Puesto que la solución particular toma la forma:
; ′′′′ ′′ ′′ ′′ sisi n 1 sin 21 cos 1 tan 1 1 cos sin 2cos 2cos cos sin 2cos cos sin 2cos coss sin 12 cos tan Resolviendo se obtiene:
;
Luego:
La solución general de la ecuación diferencial es:
15.
SOLUCION:
El polinomio característico característico de la ecuación deferencial es:
1 0
De donde: r = ± i
Con raíces imaginarias, las soluciones son: cos(x) y sen(x) , luego la solución complementaria:
= cos sin ′ ′ scoisnn csoins 1 ; ′′′′ ′′ ;
Determinamos la solución particular. Para ello determinamos el wronskiano:
Puesto que la solución particular toma la forma:
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Luego:
si n t a n 1 1 1 cos cos seccos seccosl l nsectan sectan sin cost1 an sisin n cocos cos sincoslnsectan sectan − sin− Luego la solución genera de la ecuación diferencial es:
16.
SOLUCION:
El polinomio característico característico de la ecuación deferencial es:
1 0
De donde: r = ± i
− −− 112 − − ; ; sin ′′ ′′ Con raíces reales imaginarias, las soluciones son:
y
, luego la solución complementaria complementaria
es:
Determinamos la solución particular. Para ello determinamos el wronskiano:
Puesto que la solución particular toma la forma:
Resolviendo se obtiene:
;
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− − − s i n cos 2 2 − − − − − s i n s i n 2 2 → 3 → si n cos 2 cos 32 sin2sin → → si n cos 2 cos 32 sin3cos3sin − 2 cos− 32 − sin− 3− cos− 3sin− − − cos − 2 −2 coscos− 32 − sin− 3cos 3cos− 3 sin− 3 2− Luego la solución general de la ecuación diferencial: diferencial:
17.
SOLUCION:
El polinomio característico de la ecuación diferencial es:
320 → 2 1 0 → 1 ; 2
Con raíces reales repetidas, las soluciones son: es:
y
, luego la solución complementaria complementaria
Determinamos la solución particular. Para ello determinamos el wroskiano:
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; ; cos− ′′ ′′ − cos − cos− sin− − cos − cos− − − → − ccoss sisinn coss − sin− cos− − − − − −sin − sin cos cos Resolviendo se obtiene:
;
Luego:
Hacemos
Luego la solución general de la ecuación diferencial: diferencial:
18.
SOLUCION:
El polinomio característico característico de la ecuación deferencial es:
1 0
→ − −− 112 ; ; ; De donde: r = ± i
Determinamos la solución particular. Para ello determinamos el wronskiano:
Puesto que la solución particular toma la forma:
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2 14 1cos2 1cos2 14 14 cos2 14 14 cos2 1 4 20 cos22si cos22sin22 − 2 14 1cos2 1cos2 1 1 1 4 4 cos2 4 20 cos22si cos22sin22 − 12 101 cos2 Luego la solución general de la ecuación diferencial: diferencial:
19.
SOLUCION:
El polinomio característico característico de la ecuación deferencial es:
1 0
→ − −− 112
De donde: r = ± i
Determinamos la solución particular. Para ello determinamos el wronskiano:
Puesto que la solución particular toma la forma:
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Integramos por partes:
− − − → ; 1 1 − − − 2 1 2 1 − 1 − 2 − 1 2 2 → ; ∫ 1 1 + 2 1 2 1 + 1 + 2 + Integramos por partes:
Luego la solución general de la ecuación diferencial diferencial es:
− cos 20.
SOLUCION:
El polinomio característico característico de la ecuación deferencial es:
1 0
De donde: r = ± i
raíces imaginarias
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Resolviendo se obtiene:
′′ ′′ coscos 1 1 sin2 → ; 2 2 si n 2 si n 2 2 4 4 8 4 si4n2 sin82 cossi1 n → si n cos; 2 1cos2 2 2 2 12 1cos2 2 2 14 sin22 cos si n 2cos si n 2cos cos sin 4 4 4 2 si4n x y xy y0 ;
Luego:
Luego la solución general de la ecuación diferencial diferencial es:
1.
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ddty y 0,0, ecuaci ecuación diferenciencialal homogé homogéneanea de coeficientes constantantes.es. Sea P r 10 r 1, r 1,− Luego la solsolución es: y C C , dede donde donde y Cx Cx xy 2xy 2y0 dy dy d y d − − − Sea x tlnx,además dx dt ; dx dty dydt Reemplazando en la ecuación diferencial se tiene: − ddty dydt2− dydt 2y0,simplificando ddty dydt 2y2y 0,0, ecuaci ecuación diferenci rencialal homogén homogéneaea de coeficientntes const constantntes. Sea P r r20 r 2,− r 1,1, Luego la−solsolución es: y C C , dede dond dondee y Cx Cx x y xy 9y0 dy dy d y d − − − Sea x tlnx,además dx dt ; dx dty dydt
2.
SOLUCION:
3.
SOLUCION:
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y la solucióngeneral de esta ecuación es: y C cos tC sen t y Ccos3tCsen3t, de dodond e: y C coscos3l3lnxx Csenn3lnxx 4x y 8xy 9y0 dy dy d y d − − − Sea x tlnx,además dx dt ; dx dty dydt Reemplazando en la ecuación diferencial se tiene: 4− ddty dydt8− dydt 9y0,simplificando d4 dty 12 dydt 9y0 Sea P 4r 12r90 r 32 , r 32, Luego la so⁄lución es:⁄ y C Ct , de donde:⁄ ⁄ y Cx C x lnx x y 2xy 7y0
3.
SOLUCION:
5.
SOLUCION:
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luego la solución es y C cos tC sen t y C cos√ cos√ 3t3tC sen√ n√ 3t3t, de dodon d e: y C x cos√ 3l3lnxCx sen√ n√ 3l3lnx x y 2 x y 17xy 7y0 dy dy d y d − − − Sea x tlnx,además dx dt ; dx dty dydt ddty − ddty 3 ddty 2 dydt Reemplazando en la ecuación diferencial se tiene: − ddty 3 ddty 2 dydt2− ddty dydt17− dydt 7y0 simplificando ddty 5 ddty 13 dydt 7y0
6.
SOLUCION:
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ddty 2 dydt 3y0 Sea P r 2r30 r 1, r−3, Luego la solsolución es: y −C C , dede dond dondee y Cx2C 2 2x1y 22x1 2 x1y 4y0 Sea 2 x 1 tlnx 2 ; además dydx 2− dydt ; dxd y 4− ddty dydt Reemplazando en la ecuación diferencial se tiene: . 44− ddty dydt2. 22− dydt 4y0,simplificando d4 dty 8 dydt 4y0 Sea P 4r 8r40 r 1, r 1,1,
8.
SOLUCION:
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Luego la solsoluci−ón es: y C−−C−,de donde y Cx1 C1 x2 y 5x 2y 8y0 Sea x 2 tln tlnx 2 ; además dydx − dydt ; dxd y − ddty dydt Reemplazando en la ecuación diferencial se tiene: . − ddty dydt5. − dydt 8y0,simplificando ddty 4 dydt 8y0 Sea P r 4r180 r−22i , r−22i, Luego la solsoluci−ón es: y C cos2tC−− sen2t,de donde y Cx2 cos2l cos 2l n x 2 C x 2 sen2lnx 2 6l6 l n 10.
SOLUCION:
11.
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2 6 22 2 6 2226 ; 12 ; 72 2 72; : cos sin 2 72 cosln sinln l n 7 9 → ln − ; − − − 9 1 Como
;
entonces
12.
SOLUCION: Sea
Reemplazando en la ecuación diferencial se tiene:
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Simplificando:
61 91 1 6 ; 9 , 6 19 ; − 6 19 − 6ln 19 4 22ln → ln − ; − . − − 22 14.
SOLUCION: Sea
Reemplazando en la ecuación diferencial se tiene:
Simplificando:
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316l16ln− − − → ln ; ; . − − 316− 2 316 − 230, 3 ; 1 15.
SOLUCION: Sea
Reemplazando en la ecuación diferencial se tiene:
Simplificando:
Luego la solución complementaria complementaria es:
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Simplificando:
9 sisin3n3 …. 9 9 0 ; 3 ; 3 cos3sin3 ; cos3 cos 3 sin 33 cos3 cos 3 sin 33 33cos3sin33 3cos3si cos3sin3 3cos3si cos3sin3 9cos3si cos3sin3 Sea:
La solución complementaria es:
Reemplazando en
:
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. − 4− 22 12 22 12 320, 2 ; 1 − −; : 2 2 …..α
Luego la solución complementaria complementaria es:
Reemplazando en α:
2 32 2 2 2…….
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44 4 4 0 ; 2 ; 2 ; ∶ ∶ ; ; 0 044 41 → ; 440 → 1/4 ;
Luego la solución complementaria complementaria es:
Reemplazando en:
Entonces:
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1 : ; ; ∶ 1 1 1 l n 2 23 2l n − − → ln : ; . − − 23 2 20.
SOLUCION:
Reemplazando en la ecuación diferencial se tiene:
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ln n 32 PRACTICA 9 V.Ecuación lineal homogénea: Resolver:
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2 560 0 ó:: 2, 2560 1, 3 ó−: −
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=4, 1 ó: − − − 210 0 7.
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ó−: 33 25 0 12, 12 11.
SOLUCION:
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=1, 2 ó : 9 24 160 0 15.
SOLUCION:
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2 3 1 44 ..
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i) Hacemos
2⟶2⟶
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3. Resolver
1 2
mediante una serie que satisfaga la condición
cuando
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Suponiendo que la solución es de la forma:
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SOLUCION:
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