temas de la unidad 2 de matematicas para ingenieriaDescripción completa
Full description
Laplace Transform
Descripción: ejercicios laplace
Descripción: ecuaciones diferenciales laplace
Ecuaciones DiferencialesDescripción completa
lapFull description
Ecuaciones DiferencialesDescripción completa
ejercicios resueltos de la transformada de laplace.Descripción completa
LAplace Transforms
laplace
Table of Laplace of Laplace Transforms
f ( t ) = L
-1
{F ( s )}
1
1.
1
3.
t n , n = 1, 2, 3,
7. 9.
11.
s n!
K
5.
s n +1 p
t
2 s a
sin ( at )
3 2
s + a 2 2
( s
sin ( at ) - at cos ( at )
+ a2 )
2
cos ( at ) - at sin ( at )
15.
sin ( at + b )
17.
sinh ( at )
19.
e
at
21.
e
at
23.
t ne at , n = 1, 2, 3,
25.
uc ( t ) = u ( t - c )
2
2
( s - a ) + b 2
( s - a ) - b2
1
f ( t ) t
33.
ò f ( t - t ) g (t ) d t
35.
f ¢ ( t )
37.
f
( t )
2
b
f ( t )
( n)
2
2
ct
0
2
s 2 - a 2 b
n!
( s - a ) e
e
n +1
t , p > -1
6.
t
8.
cos ( at )
10.
t cos ( at )
- cs
F ( s - c )
ò
12.
, n = 1, 2, 3,
1× 3 × 5
L
K
F ( u ) du
( 2n - 1)
2n s s
s 2 + a 2 s 2 - a 2
( s
+ a2 )
2
16.
cos ( at + b )
18.
cosh ( at )
20.
e
at
22.
e
at
24.
f ( ct ) d
2
( s + a ) s ( s + 3a ) ( s + a ) 2
2
2
2
2
2
s cos ( b ) - a sin ( b ) s 2 + a 2 s s 2 - a 2 s - a
cos ( bt )
2
( s - a ) + b 2 s - a
cosh ( bt )
2
( s - a ) - b2 1 æsö F ç ÷ c ècø
(t - c)
e
- cs
Dirac Delta Function
28.
uc ( t ) g ( t )
30.
t n f ( t ) , n = 1, 2, 3,
32.
2
2as 2
sin ( at ) + at cos ( at )
cos ( at ) + at sin ( at )
26.
p
n + 12
14.
¥
s
n - 12
s p +1
2
- cs
s F ( s )
Heaviside Function
t
2
s + a a
K
e
31.
2
2
sinh ( bt )
29.
4.
s - a G ( p + 1)
2
s sin ( b ) + a cos ( b )
sin ( bt )
u c (t ) f ( t - c )
2
( s + a ) s ( s - a ) ( s + a )
1
e
2a 3
2
13.
2
F ( s ) = L { f ( t )}
at
2.
2as
t sin ( at )
27.
f ( t ) = L -1 {F ( s )}
F ( s ) = L { f ( t )}
e K
ò
t 0
f ( v ) dv
L
{ g ( t + c )} n
( -1) F ( n) ( s ) F ( s ) s
F ( s ) G ( s )
34.
f ( t + T ) = f ( t )
sF ( s ) - f ( 0 )
36.
f ¢¢ ( t )
s n F ( s ) - s n -1 f ( 0 ) - s n - 2 f ¢ ( 0 )
- cs
L
ò
T 0
e
- st
f ( t ) dt
1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ¢ ( 0 )
- sf ( n- 2) ( 0 ) - f ( n-1) ( 0 )
Table Notes 1. This list is not a complete listing of Laplace transforms and only contains some o f the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions. cosh ( t ) =
e
t
+ e- t
sinh ( t ) =
2
e
t
- e-t
2
3. Be careful when using “normal” trig function vs. hyperbolic hyperbolic functions. The only 2 difference in the formulas is the “+ a ” for the “normal” trig functions becomes a 2 “- a ” for the hyperbolic functions! 4. Formula #4 uses the Gamma function which is defined as
G (t ) = ò
¥
e
- x t -1
0
x
dx
If n If n is a positive integer then,
G ( n + 1) = n ! The Gamma function is an extension of the normal factorial factorial function. Here are a couple of quick facts for the Gamma function
