Example 12.33. Use convolution to solve the initial value problem

              
            with
            .  

                         
                                    A graph of the solution.

Explore Solution 12.33.

The solution is accomplished in two steps. (a) First solve    with  .  
(b)  Second solve for    with  .  Then the desired solution is  .  

(a) First solve    with  .  

(b)  Second solve for    with  .
The convolution theorem is used to make the construction of    where    and  .  

Aside. We can substitute v[t] into the D.E.'s and verify that it is the solution.

Remark. Mathematica 4 obtained a different form of the solution and the following comment applies when using it.
Now guide Mathematica using  trigonometric substitutions to simplify the expression for  v[t].
Notice.  This cannot be done automatically.

Aside. We can substitute y[t] into the D.E.'s and verify that it is the solution.

Aside. We use Mathematica/s DSolve package to solve the D.E. and plot the solution.

Remark. This involves the version unsimplified of v[t].  So we will plot our previously simplified function y[t].

(c) 2006 John H. Mathews, Russell W. Howell