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Find the Inverse Laplace transforms of the following functions:   (4s + 2)/(s2 +6s + 34) 1/ [(s-2)(s-4)(s-8)]

Find the Inverse Laplace transforms of the following functions:   (4s + 2)/(s2 +6s + 34) 1/ [(s-2)(s-4)(s-8)]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the Inverse Laplace transforms of the following functions:

 

  1. (4s + 2)/(s2 +6s + 34)
  2. 1/ [(s-2)(s-4)(s-8)]

Subject : DIFFERENTIAL EQUATION
Topic: Inverse Laplace Transforms

Below is the Table of Laplace Transforms.

 

1. 3. 5. 7. 9. f(t)=2²¹{F(s)} 1 19. t", n=1,2,3,... 11. sin(at)-at cos(at) 21. √t sin(at) t sin (at) 13. cos(at)-at sin (at) 15. sin(at+b) 17. sinh(ar) et sin (br) e” sinh(br) 23. te, n=1,2,3,... u(t)= u(t-c) Heaviside Function 25. 27. u(t)f(t-c) 29. eºf(t) 31. ƒ(1) 33. ff(t-1)g(1) dr 35. f'(t) 37. f(t) Table of Laplace Transforms F(s) = {f(t)} S L su+1 a s² + a² 2as (s² + a²)² 2a³ (s² + a²)²³ s(s²_a²) (s² + a²)² s sin (b) + acos (b) s² + a² 3²-d² b (s-a)² + b² b (s-a)²-B² n! (s-a)*** S e¯ F (s) F(s-c) *F(u) du 2. f(t)=2²¹{F(s)} at 4. tº,p>-1 e 6., n=1,2,3,... 8. cos(at) 10. tcos(at) 20. 12. sin (at) + at cos(at) 22. 14. cos(at)+ at sin(at) 16. cos(at+b) 18. cosh (at) e cos (bt) ea cosh (br) 24. f(ct) 8(t-c) Dirac Delta Function u. (t)g(t) 26. 28. 30. tf(t), n=1,2,3,... 32. ff(v) dv F(s) G(s) SF (s)-f(0) 36. f(t) 34. f(t+T)=f(t) F(s) = £{f(t)} 1 s-a [(p+1) Sit 1-3-5---(2n-1)√√ 2" 5" 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 3-0² s-a (s-a)² + b² s-a (s-a)²³-5² 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)-5-¹ƒ (0)-5-²ƒ' (0)---sf("~²) (0) — ƒ(¹)(0) -
Transcribed Image Text:1. 3. 5. 7. 9. f(t)=2²¹{F(s)} 1 19. t", n=1,2,3,... 11. sin(at)-at cos(at) 21. √t sin(at) t sin (at) 13. cos(at)-at sin (at) 15. sin(at+b) 17. sinh(ar) et sin (br) e” sinh(br) 23. te, n=1,2,3,... u(t)= u(t-c) Heaviside Function 25. 27. u(t)f(t-c) 29. eºf(t) 31. ƒ(1) 33. ff(t-1)g(1) dr 35. f'(t) 37. f(t) Table of Laplace Transforms F(s) = {f(t)} S L su+1 a s² + a² 2as (s² + a²)² 2a³ (s² + a²)²³ s(s²_a²) (s² + a²)² s sin (b) + acos (b) s² + a² 3²-d² b (s-a)² + b² b (s-a)²-B² n! (s-a)*** S e¯ F (s) F(s-c) *F(u) du 2. f(t)=2²¹{F(s)} at 4. tº,p>-1 e 6., n=1,2,3,... 8. cos(at) 10. tcos(at) 20. 12. sin (at) + at cos(at) 22. 14. cos(at)+ at sin(at) 16. cos(at+b) 18. cosh (at) e cos (bt) ea cosh (br) 24. f(ct) 8(t-c) Dirac Delta Function u. (t)g(t) 26. 28. 30. tf(t), n=1,2,3,... 32. ff(v) dv F(s) G(s) SF (s)-f(0) 36. f(t) 34. f(t+T)=f(t) F(s) = £{f(t)} 1 s-a [(p+1) Sit 1-3-5---(2n-1)√√ 2" 5" 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 3-0² s-a (s-a)² + b² s-a (s-a)²³-5² 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)-5-¹ƒ (0)-5-²ƒ' (0)---sf("~²) (0) — ƒ(¹)(0) -
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 ( 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! 4. Formula #4 uses the Gamma function which is defined as r(t)=x²²dx 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 I(p+1)=pr (p) p(p+1)(p+2).(p+n−1)=- (2)=√ I(p+n) I(p)
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 ( 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! 4. Formula #4 uses the Gamma function which is defined as r(t)=x²²dx 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 I(p+1)=pr (p) p(p+1)(p+2).(p+n−1)=- (2)=√ I(p+n) I(p)
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