9.4.1: The First Shifting Theorem (Exercises)

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In Exercises 1-5 find the Laplace transform of the function.

1. \(te^{10t}\)

2. \(t^3e^{-2t}\)

3. \(e^t\sin 3t\)

4. \(e^{6t}(1+3t-t^2)\)

5. \((1+e^t-3e^{-2t})\cosh 4t\)

In Exercises 6-34 find the inverse Laplace transform of the function.

6. \({3\over(s-7)^4}\)

7. \({2s-4\over s^2-4s+13}\)

8. \({1\over s^2+4s+20}\)

9. \({1\over(s-2)^2-4}\)

10. \({2\over(s-3)^2-9}\)

11. \({2s+3\over(s-7)^4}\)

12. \({s^2-1\over(s-2)^6}\)

13. \({s+5\over s^2+6s+18}\)

14. \( {s\over s^2+2s+1}\)

15. \( {s^3+2s^2-s-3\over(s+1)^4}\)

16. \( {2s+3\over(s-1)^2+4}\)

17. \( {3\over(s+2)^2}-{2s+6\over s^2+4}\)

18. \( {3s^2+2s+1\over(s^2+1)(s^2+2s+2)}\)

19. \( {3s+2\over(s-2)(s^2+2s+5)}\)

20. \( {17 s-15\over(s^2-2s+5)(s^2+2s+10)}\)

21. \( {8s+56\over(s^2-6s+13)(s^2+2s+5)}\)

22. \( {s+9\over(s^2+4s+5)(s^2-4s+13)}\)

23. \( {3s-2\over(s^2-4s+5)(s^2-6s+13)}\)

24. \( {3s-1\over(s^2-2s+2)(s^2+2s+5)}\)

25. \( {20s+40\over(4s^2-4s+5)(4s^2+4s+5)}\)

26. \( {1\over(s-1)(s^2-2s+17)}\)

27. \( {3s+2\over(s-2)(s^2+2s+10)}\)

28. \( {34-17s\over(2s-1)(s^2-2s+5)}\)

29. \( {s+2\over(s-3)(s^2+2s+5)}\)

30. \( {2s-2\over(s-2)(s^2+2s+10)}\)

31. \( {s+2\over(s^2+2s+2)(s^2-1)}\)

32. \( {2s-1\over(s^2-2s+2)(s+1)(s-2)}\)

33. \( {2s-3\over s(s-2)(s^2-2s+5)}\)

34. \( {5s-15\over(s^2-4s+13)(s-2)(s-1)}\)

In Exercises 35-49 solve the initial value problem using Laplace transforms.

35. \(y'+4y=e^{-4t}, \quad y(0)=2,\)

36. \(y''+2y'+y=0, \quad y(0)=1,\; y'(0)=1\)

37. \(y''-6y'+9y=t, \quad y(0)=0,\; y'(0)=1\)

38. \(y''-6y'+13y=0, \quad y(0)=0,\; y'(0)=-3\)

39. \(y''-y'=e^t\cos t, \quad y(0)=0,\; y'(0)=0\)

40. \(y''+2y'+y=6\sin t-4\cos t, \quad y(0)=-1,\; y'(0)=1\)

41. \(y''-5y'+6y=10e^t\cos t, \quad y(0)=2,\quad y'(0)=1\)

42. \(y''+2y'+2y=2t, \quad y(0)=2,\quad y'(0)=-7\)

43. \(y''-2y'+2y=5\sin t+10\cos t, \quad y(0)=1,\; y'(0)=2\)

44. \(y''+4y'+13y=10e^{-t}-36e^t, \quad y(0)=0,\; y'(0)=-16\)

45. \(y''+4y'+5y=e^{-t}(\cos t+3\sin t), \quad y(0)=0,\quad y'(0)=4\)

46. \(2y''+2y'+y=2t, \quad y(0)=1,\; y'(0)=-1\)

47. \(4y''-4y'+5y=4\sin t-4\cos t, \quad y(0)=0,\; y'(0)=11/17\)

48. \(4y''+4y'+y=3\sin t+\cos t, \quad y(0)=2,\; y'(0)=-1\)

49. \(9y''+6y'+y=3e^{3t}, \quad y(0)=0,\; y'(0)=-3\)

50. A 4 pound weight stretches a spring 2 feet. The weight is released from rest 18 inches above equilibrium, and the motion takes place in a medium offering a damping force equal to \(7/8\) times the instantaneous velocity. Use Laplace transforms to find the equation of motion.

51. Given that \(f(t)\leftrightarrow F(s)\), find the inverse Laplace transform of \(F(as-b)\), where \(a>0\).

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