12 .10 • CONVOL UTION 579- Find .C (J; e-T cos (t - r) dr).
16. Find .C (l; (t - r)^2 e^7 dr).
- Let F(s) = .C(f(t)). Use convolution to show that .c-^1 ( F;s)) = l; f(r)dr.
For Exercises 18-21, use the convolution theorem t o solve t he integral equation.
- /(t)+4J;(t-r)/(r)dr=2.
19. f (t) = e' + l; e•-T f (r) dr.
- f(t) = 2t+ l;sin(t- r)f(r)dr.
- 6/ (t) = 2t^3 + l; (t - r)^3 f (r) dr.
For Exercises 22- 25, solve the initial value problem.
- y" (t) - 2y' (t) + 5y (t) = 20 (t), with y (0) = 0 and y' {O) = 0.
23. y" (t) + 2y' (t) + y (t) = o (t), with y (0) = 0 and y' (0) = O.
- y" (t) +4y'(t) +3y(t) = 2o(t), with y(O) = 0 and y' (0) = 0.
- y" (t) + 4y' (t) + 3y (t) = 28 (t -1), with y (0) = 0 and y' (0) = 0.
For Exercises 26-29, use the IVP convolution method t o solve the initial value
problem.- y" (t) - 2y' (t) + 5y (t) = 8exp (-t), with y (0) = 1 and y' (0) = 2.
27. y" (t) + 2y' (t) + y (t) = t^4 , with y (0) = l and y' (0) = 2.
- y" (t) + 4y' (t) + 3y (t) = 24t^2 e- •, with y (O) = l and y' (0) = 2.