Background. The forced motion of a mechanical system satisfies the nonhomogeneous linear differential equation . We investigate the solution when F(t) is a Fourier series.

Example 1. Find the general solution to ,
where , extended periodically with period .
Recall that .

First, set up the n-th term for F(t) and U(t).

Substitute the n-th terms in the D.E. and solve for A[n] and B[n].

Form the n-th term for U[t] and substitute it in the D.E.

Form partial sum S[t] of 5 terms of U[t] and plot it.

Example 2. Find the general solution to , where , extended periodically with period . Recall that we know that .

First, set up the n-th term for F(t) and U(t).

Substitute the n-th terms in the D.E. and solve for A[n] and B[n].

Form the n-th term for U[t] and then form partial sum S[t] of 5 terms of U[t] and plot it.

(c) John H. Mathews, 1998