1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

57 8 C HAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

Therefore, the solution is

cost

y(t) =u(t)+v(t) = cost+3sint+cos(t)ln

1

. t
+sm

-------..-EXERCISES FOR SECTION 12 .10

For Exercises 1-4, find the indicated convolution.

1. t * t


2. tSint
   3. e^1 e^21
   4. sin h sin 2t

For Exercises 5-8, use convolution to find £,-^1 (F (s )).

2

(^5) · F(s) = (s - l)(s -2)'
6

6. F(s) = sa·
   1


7. F (s) = s (s2 + l)'
   8


8. F(s) = (s2 + l)(s2 +4)'


9. Prove the distributive law for convolution, f (g + h) = f g + f * g.


10. Use the convolution t heorem and mathematical induction to show that

.c-• ( l. ) = l t"-•e•'.

(s -at (n - l)!

11. Find .c-^1 (-
    8
    s - 1 -) •


12. Find c,-^1 (s/: 1 ).


13. Use the convolution theorem to solve the initial value problem

y" (t) + y (t) = 2 sin t, with y (0) = 0 and y' (0) = O.

1 4. Use the convolution theorem to show that the solution to the initial value problem

y" (t) + w^2 y (t) = f (t), with y (0) = 0 and y' (0) = 0, is

y(t) = -11' f (7)sin[w (t -T)}dT.

w 0