Find the Laplace transforms of the following functions: 1. x2 2. xe6x Subject : DIFFERENTIAL EQUATION Topic: Laplace Transforms Below is the Table of Laplace Transforms
Find the Laplace transforms of the following functions: 1. x2 2. xe6x Subject : DIFFERENTIAL EQUATION Topic: Laplace Transforms Below is the Table of Laplace Transforms
Related questions
Question
Find the Laplace transforms of the following functions:
1. x2
2. xe6x
Subject : DIFFERENTIAL EQUATION
Topic: Laplace Transforms
Below is the Table of Laplace Transforms.
Transcribed Image Text:1.
3.
5.
7.
9.
f(t)=2¹{F(s)} F(s)=L{ƒ(t)}
1
t", n=1,2,3,...
√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)
19. e sin (br)
21. esinh(br)
23. te, n=1,2,3,...
u(t)= u(t-c)
Heaviside Function
25.
27. u(t)f(t-c)
29. ef(t)
31. ƒ(1)
Table of Laplace Transforms
33. ff(t-1)g(t)dt
35. f'(t)
37. f(") (1)
S
a
s² + a²
2as
(s² + a²)²
2a³
(s² + a²)²
s(s²-a²)
(s² + a²)²
s sin (b) + acos (b)
s² + a²
(s-a)² +3²
b
(s-a)²-b²
n!
(s-a) ***
S
e™ F (s)
F(s-c)
*F(u) du
ƒ(1)=L²¹{F(s)}
2. eª
4. tº,p>-1
6. -1,2,3,...
8. cos(at)
10. tcos(at)
12. sin(at)+ at cos(at)
14. cos(at)+ at sin(at)
16. cos(at+b)
18. cosh(at)
20. e" cos(bt)
22. e cosh (br)
24. f(ct)
8(t-c)
Dirac Delta Function
26.
28. u(t)g(t)
30. "f(t), n=1,2,3,...
32. ff(v) dv
34. f(t+T)=f(t)
F(s) G(s)
SF (s)- ƒ (0) 36. f"(t)
F(s)=L{f(t)}
s-a
I(p+1)
1-3-5---(2n-1)√
2"+
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
S
3²-a²
s-a
(s-a)² +3²
s-a
(s-a)²-b²
풍미용)
e{g(t+c)}
(-1)" F") (s)
F(s)
S
fe" f(t) at
1-e-
s²F (s)-sf (0)-f'(0)
s" F (s)-s¹ƒ(0)-5-²ƒ' (0) --- sf(-²) (0) — ƒ(¹) (0)
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.
e' + e
cosh (1)
2
sinh (t)=z
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)=x²²x
exdx
If n is a positive integer then.
I(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
[(p+1)=pr (p)
p(p+1)(p+2)...(p+n-1)=
=√√π
I(p+n)
Γ(p)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
