Find the Laplace transforms of the following functions: 1. 3(x-1) + e-x 2. 4 sin(3x) + 2 cos(9x)
Find the Laplace transforms of the following functions: 1. 3(x-1) + e-x 2. 4 sin(3x) + 2 cos(9x)
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Question
Find the Laplace transforms of the following functions:
1. 3(x-1) + e-x
2. 4 sin(3x) + 2 cos(9x)
Subject : DIFFERENTIAL EQUATION
Topic: Laplace Transforms
Below is the Table of Laplace Transforms.
Transcribed Image Text:Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of
the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions.
cosh (t)=
e'+e
2
-e
sinh()-zễ
2
3. Be careful when using "normal" trig function vs. hyperbolic functions. The only
difference in the formulas is the "+ a²" for the "normal" trig functions becomes a
"-a²" for the hyperbolic functions!
4. Formula #4 uses the Gamma function which is defined as
r(t)=√³x²¹x
If n is a positive integer then,
T(n+1)=n!
The Gamma function is an extension of the normal factorial function. Here are a
couple of quick facts for the Gamma function
T(p+1)=pr (p)
p(p+1) (p+2).(p+n−1)=-
I(+)-√F
(p+n)
Γ(p)
Transcribed Image Text:1.
3.
5.
7.
9.
f(t)=2²¹{F(s)} F(s) = £{f(t)}
1
t", n=1,2,3,...
19.
√t
sin(at)
t sin (at)
11. sin(at) at cos(at)
13. cos(at)-at sin (at)
15. sin(at+b)
17.
sinh(at)
sin (br)
21. e” sinh(br)
ee
23. te, n=1,2,3,...
u(t)= u(t-c)
Heaviside Function
25.
27. u(t)f(t-c)
29. eªf(t)
31. ƒ(1)
Table of Laplace Transforms
33. ff(t-t)g(t)dt
35. f'(t)
37. f(") (1)
ESTE
a
s² + a²
2as
(s² + a²)²
2a³
(s² + a²)²
s(s²-a²)
(s² + a²)²
s sin (b) + a cos (b)
s² + a²
a
s²-a²
(s-a)² + b²
b
(s-a)²-5²
n!
(s-a)***
S
e™ F (s)
F(s-c)
*F(u) du
2.
4. tº.p>-1
f(t)=2²¹{F(s)}
6., n=1,2,3,...
at
8. cos(at)
10. tcos(at)
12. sin(at)+ at cos(at)
20.
14. cos(at)+ at sin(at)
16. cos(at+b)
18.
cosh (at)
22.
26.
cos (bt)
e" cosh (br)
24. f(ct)
8 (1-c)
Dirac Delta Function
u. (t)g(t)
28.
30. "f(t), n=1,2,3,...
32. f f (v) av
F(s) G(s)
SF (s)-f(0) 36. f(t)
34. f(t+1)=f(t)
F(s) = £{f(t)}
1
s-a
(p+1)
5 +1
1-3-5---(2n-1)√√
25+
S
s² + a²
s²-a²
(s² + a²)²
2as²
(s² + a²)²
s(s²+3a²)
(s² + a²)²
scos (b)-a sin (b)
s² + a²
5
s²-a²
s-a
(s-a)² + b²
s-a
(s-a)²-b²
+ F(-:-)
e¯ {g(t+c)}
(-1)" F") (s)
F(s)
S
fe f(t) dt
1-e-
s²F (s)-sf (0)-f(0)
s" F (s)-s-¹ƒ(0)-s-²ƒ' (0)---sf(-²) (0) — ƒ(¹) (0)
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