Convolution products and differential equations

Question

Assignment:

Q1)   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.

Q2) 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

Provide complete and step by step solution for the question and show calculations and use formulas.