Find the Laplace transforms of the following functions: 1. cos 4x 2. e-x sin 2x Subject : DIFFERENTIAL EQUATION Topic: Laplace Transforms Below is the Table of Laplace Transforms.
Find the Laplace transforms of the following functions: 1. cos 4x 2. e-x sin 2x Subject : DIFFERENTIAL EQUATION Topic: Laplace Transforms Below is the Table of Laplace Transforms.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the Laplace transforms of the following functions:
1. cos 4x
2. e-x sin 2x
Subject :
Topic: Laplace Transforms
Below is the Table of Laplace Transforms.
Transcribed Image Text:1.
3.
5.
7.
9.
f(t)=2²¹{F(s)} F(s)=L{ƒ(t)}
1
19.
t", n=1,2,3,...
11. sin(at) at cos(at)
21.
√i
sin(at)
tsin (at)
13. cos(at)-at sin(at)
15. sin(at+i
17. sinh(at)
e sin (br)
e“sinh(br)
23. te, n=1,2,3,...
25. u(t)=u(t-c)
Heaviside Function
27. u(t)f(t-c)
29. ef(t)
31.
1(0)
Table of Laplace Transforms
33. ff(t-1)g(1) dr
35. f'(t)
37. f(t)
編號
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
B
(s-a)² + b²
b
(s-a)²-3²
n!
(s-a)***
e-cs
S
e™ F (s)
F(s-c)
*F(u) du
2.
4.
T(1)=L²¹{F(s)}
e
6. -1,2,3,...
tº,p>-1
8. cos(at)
10. tcos(at)
12. sin(at)+ at cos(at)
20.
14. cos (at) + at sin(at)
22.
16. cos(at+b)
18.
cosh (at)
Pª
cos (bt)
cosh (br)
24. f(ct)
8(t-c)
26.
Dirac Delta Function
28.
u. (1) 8 (1)
30. "f(t), n=1,2,3,...
32. ff(v) dv
F(s) G(s)
SF (s)-f(0) 36. f(t)
34. f(t+1)=f(t)
F(s) = £{f(t)}
1
s-a
I(p+1)
5p+1
1-3-5-(2n-1)√√
2" 5"+
S
3² + a²
s²-a²
(s² + a²)²
2as²
(s² + a²)²
s(s²+3a²)
(s² + a²)²
scos (b)-asin (b)
s² + a²
S
s²-a²
s-a
(s-a)² + b²
s-a
(s-a)²-3²
+ F(-)
e{g(t+c)}
(-1)" F") (s)
F(s)
S
fef(t) dt
1-e
s²F (s)-sf (0)-f(0)
s" F (s)-5-¹ƒ(0)-5-²ƒ' (0)-sf(-²) (0)-f(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.
cosh (t)=
e²+e¹
2
sinh
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
**- a*** for the hyperbolic functions!
If n is a positive integer then
e¹ - e*
2
4. Formula #4 uses the Gamma function which is defined as
r(t)=*e*x²¹dx
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
r(p+1)=pr (p)
p(p+1)(p+2)(p+n−1)=-
T(-1)=√T
I(p+n)
Γ(p)
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