7.5E: Convolution (Exercises)

In Exercises 1-27 find the Laplace transform of the given function.

1. \(f(t)=t\cos 2t\)

2. \(f(t)=t^2\sinh t\)

3. \(f(t)=te^{2t}\sin 6t\)

4. \(f(t)=t\sinh 3t\)

5. \(f(t)=t^2\cos t\)

6. \(f(t)=te^{-3t}\cos 3t\)

7. \(\int_0^te^u\, du\)

8. \(\int_0^te^{-u}\cos u\, du\)

9. \(\int_0^tue^{t-u}\, du\)

10. \(t\int_0^t\sin u\, du\)

11. \(\int_0^t\cos u\, du\)

12. \(\int_0^t u\sin u\, du\)

13. \(\int_0^t\sin u \cos b(t-u)\, du\) 

14. \(t\int_0^tue^{-u}\, du\)

15. \(\int_0^t\sin au\cos b(t-u)\, du\)

16. \(\int_0^t e^u\sin a(t-u)\,du\)

17. \(\int_0^t\sinh au\cosh a(t-u)\,du\)

18. \(e^t\int_0^t\sin\omega u \cos\omega (t-u)\,du\)

19. \(e^t\int_0^tu^2 (t-u)e^u\,du\)

20. \(e^{-t}\int_0^t e^{-u}u\cos\omega (t-u)\,du\)

21. \(e^t\int_0^t e^{2u}\sinh (t-u)\,du\)

22. \(\int_0^tu e^{2u}\sin 2(t-u)\,du\)

23. \(\int_0^t (t-u)^3 e^u\, du\)

24. \(\int_0^tu^6 e^{-(t-u)}\sin 3(t-u)\,du\)

25. \(\int_0^tu^2 (t-u)^3\, du\)

26. \(\int_0^t (t-u)^7 e^{-u} \sin 2u\,du\)

27. \(\int_0^t (t-u)^4\sin 2u\,du\)

In Exercises 28-31 find the inverse Laplace transform of the given function.

28. \({4s\over(s^2+1)^2}\)

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

30. \( {12s-24\over(s^2-4s+85)^2}\)

31. \( {s^2-4s+3\over(s^2-4s+5)^2}\)

In Exercises 32-45 express the inverse transform as an integral.

32. \(1\over s^2(s^2+4)\)

33. \(s\over(s+2)(s^2+9)\)

34. \(s\over(s^2+4)(s^2+9)\)

35. \(s\over(s^2+1)^2\)

36. \(1\over s(s-a)\)

37. \(1\over(s+1)(s^2+2s+2)\)

38. \(1\over (s+1)^2(s^2+4s+5)\)

39. \(1\over(s-1)^3(s+2)^2\)

40. \(s-1\over s^2(s^2-2s+2)\)

41. \(s(s+3)\over(s^2+4)(s^2+6s+10)\)

42. \(1\over(s-3)^5s^6\)

43. \(1\over(s-1)^3(s^2+4)\)

44. \(1\over s^2(s-2)^3\)

45. \(1\over s^7(s-2)^6\)

In Exercises 46-53 solve the initial value problem using Laplace transforms.

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

47. \(y''+4y=8\sin2t+9\cos t, \quad y(0)=1,\quad y'(0)=0\)

48. \(y'+y=t\sin t, \quad y(0)=0\)

49. \(y''+9y=\cos 3t, \quad y(0)=2,\quad y'(0)=5\)

50. \(y''+16y=\left\{\begin{array}{cl} {\cos 4t,}&{0 \le t<\pi,}\\ {0,} & {t\ge\pi.} \end{array}\right. , \quad y(0)=0,\quad y'(0)=1\)

51. \(y'-y=te^t\sin t, \quad y(0)=0\)

52. \(y''+y=\sin t, \quad y(0)=1,\quad y'(0)=-1\)

53. \(y''+y=\left\{\begin{array}{cl} {1,}&{0 \le t<\pi/2,}\\ {\sin t,} & {t\ge\pi/2.} \end{array}\right. , \quad y(0)=1,\quad y'(0)=0\)

In Exercises 54-69 solve the integral equation.

54. \(y(t)+\int_0^t(t-u) y(u)\,du=t\)

55. \(y(t)=2t-4\int_0^t\sin u y(t-u)\,du\)

56. \(y(t)=te^t+\int_0^ t uy(t-u)\,du\)

57. \(y(t)+2\int_0^t y(u) \cos (t-u)\,du=4e^{-t}+\sin t\)

58. \(y(t)+\int_0^ty(u)\,du=1\)

59. \(y(t)=\cos t+\int_0^te^{-u}y(t-u)\,du\)

60. \(y(t)=1+t-{8\over 3}\int_0^t(u-t)^3y(u)\,du\)

61. \(t-2y(t)=\int_0^t(e^u-e^{-u})y(t-u)\,du\)

62. \(y'(t)=1-\sin t- \int_0^t y (u)\,du,\quad y(0)=0\)

63. \(y'(t)+6y(t)+9\int_0^ty (u)\,du=1,\quad y(0)=0\)

64. \(y(t)=t-\int_0^t (t-u) y(u)\,du\)

65. \(y(t)=\sin t-2 \int_0^t\cos (t-u) y (u)\,du\)

66. \(y(t)=1+2 \int_0^ty(u)\cos(t-u)\,du\)

67. \(y(t)=t+\int_0^t y(u)e^{-(t-u)}\,du\)

68. \(y'(t)=t+\int_0^t y(u)\cos (t-u)\,du,\, y(0)=4\)

69. \(y(t)=\cos t-\sin t+ \int_0^t y(u)\sin (t-u)\,du\)

In Exercises 70-75 find the Laplace transforms of the given periodic functions.

70. \( {f(t)=\left\{\begin{array}{cl} 1, & 0\le t<a,\\ -1, & a\le t<2a,\end{array}\right.\hskip30pt f(t+2a)=f(t), \quad t\ge 0}\)

71. \( {f(t)=\left\{\begin{array}{cl} 1, & 0\le t<a,\\ 0, & a\le t<2a,\end{array}\right.\hskip30pt f(t+2a)=f(t), \quad t\ge 0}\)

72. \( {f(t)={a\over b}t,\quad 0\le t<b. \hskip30ptf(t+b)=f(t), \quad t\ge 0}\)

73. \( {f(t)=\left\{\begin{array}{cl} t, & 0\le t<1,\\ 2-t, & 1\le t<2,\end{array}\right.\hskip30pt f(t+2)=f(t), \quad t\ge 0}\)

74. \(f(t)=|\sin t|\)

75. \( {f(t)=\left\{\begin{array}{cl} \sin t, & 0\le t<\pi,\\ 0, & \pi\le t<2\pi,\end{array}\right.\hskip30pt f(t+2\pi)=f(t), \quad t\ge 0}\)

76. Prove that \(f\ast g=g\ast f\).

77. Prove that if \(f(t)\leftrightarrow F(s)\) then

\[\int_0^tf(u)\,du\leftrightarrow {F(s)\over s}.\nonumber \]

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