Determine the convolution products and differential equation

Question

Laplace Transforms : Convolution Products and Differential Equations

(1) Calculate the following convolution products :-

                                                     
(a) t * e ^t                                      (b) e ^t * e ^-t

(c) sin t * sin t                          (d) H ( t - 1 ) * t²

Hint :- cos (A - B) - cos (A + B) = 2 sin A sin B.

(2) Find the solutions of the differential equations :-

     .
(a) x + x = H ( t - 1 ) x (0) = 0

     .
(b) x + x = 1 - H ( t - 1 ) x (0) = 0

     .
(c) x + x = H ( t - 1 ) - H (t - 2 ) x (0) = 0