f(t) = L−¹{F(s)}(t) 1. 1 2. e-at 3. t", n positive integer 4. sin(at) 5. cos(at) 6. sinh(at) 7. cosh(at) 8. e-at f(t) 9. e-at sin(bt) 10. e-at cos(bt) 11. teat, n positive integer 12. u(t-a), a > 0 13. u(ta)f(t-a) 1. [ f(7) dr 14. Table of Laplace transforms F(s) = L{f(t)}(s) 1 15. ·[ f(t-7)g(7)dr 16. 8(t - a) 17. f(n) (t) 18. (-t)" f(t) 8 1 s+a n! 8"+1¹ a 8>0 s² + a²: 8 8² + a²¹ a s²-a²¹ S 8²-a2¹ F(s+a) T s> -a s>0 F(s) 8 -98 b (s+a)² + b²¹ s+a (s+a)² +6² n! (s+a)n+1¹ e-as 8 e-as F(s) 8>0 F(s)G(s) 8>0 s> lal s> lal 8>0 first shifting law s> -a s-a s-a second shifting law s" F(s)-8"-1 f(0) --- f(n-¹) (0) F(n) (s)
f(t) = L−¹{F(s)}(t) 1. 1 2. e-at 3. t", n positive integer 4. sin(at) 5. cos(at) 6. sinh(at) 7. cosh(at) 8. e-at f(t) 9. e-at sin(bt) 10. e-at cos(bt) 11. teat, n positive integer 12. u(t-a), a > 0 13. u(ta)f(t-a) 1. [ f(7) dr 14. Table of Laplace transforms F(s) = L{f(t)}(s) 1 15. ·[ f(t-7)g(7)dr 16. 8(t - a) 17. f(n) (t) 18. (-t)" f(t) 8 1 s+a n! 8"+1¹ a 8>0 s² + a²: 8 8² + a²¹ a s²-a²¹ S 8²-a2¹ F(s+a) T s> -a s>0 F(s) 8 -98 b (s+a)² + b²¹ s+a (s+a)² +6² n! (s+a)n+1¹ e-as 8 e-as F(s) 8>0 F(s)G(s) 8>0 s> lal s> lal 8>0 first shifting law s> -a s-a s-a second shifting law s" F(s)-8"-1 f(0) --- f(n-¹) (0) F(n) (s)
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
Transcribed Image Text:f(t)=L-¹{F(s)}(Ⓒ)
1. 1
2. e-at
3. t", n positive integer
4. sin(at)
5. cos(at)
6. sinh(at)
7. cosh(at)
8. e-at f(t)
9. e-at sin(bt)
Table of Laplace transforms
F(s) = L{f(t)}(s)
1
Ti
8
10. eat cos(bt)
11. teat, n positive integer
12. u(ta), a > 0
13. u(ta)f(t-a)
f f(7) dr
S₁
14.
15.
f(t-r)g(r)dr
16. 8(ta)
17. f(n) (t)
18. (t)" f(t)
Je
eat cos bt dt =
J eat sin bt dt =
1
s+a
n!
sn+1¹
8
8² + a²¹
a
a
82 +92 80
8>0
8
82-92¹
F(s+a)
8
F(s)
8
e-as
s-a
b
(s+a)² + b²¹
s+a
(s+a)² +62¹
n!
(s+a)n+1¹
e-as
s>0
eat
a²+6²
F(s)G(s)
pat
a²+ b²
2
8>0
s> lal
s> |a|
8>0
first shifting law
8-a
s" F(s) - s"-f(0)
F(n) (s)
s-a
8-a
second shifting law
(a cos bt+b sin bt)
(-b cos bt + a sin bt)
f(n-1) (0)
DX
Transcribed Image Text:8. Find the Laplace transform of the solution to the initial-value problem: x" +2x' + x = f(t) and
x(0) = x'(0) = 0, with:
f(t)
=
2,
t,
if 0 ≤ t ≤ 2,
if t > 2.
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