Extra Example 1.  Find the general solution to  ,  where is given by the Fourier series  . 

Explore Extra Solution 1.

First, find the transient solution to the differential equation.

Next, use Fourier series to construct the steady state solution to the differential equation.  Enter the formula for the n-th terms T[n,t] and F[n,t] of the Fourier Series for U[t] and F[t], respectively.  Then form the equation relating the n-terms and the equations involving Cos[n t] and Sin[n t].  Then solve these equations for the Fourier coefficients of  U[t].

Use the Fourier coefficients of U[t] and form the n-th term of the series  U[n,t], and verify that it satisfies the given D.E.

Construct the Trigonometric Polynomial    of degree n = 5 and verify that it satisfies the D.E.

Look at the size of the coefficients    and observe that
    
We can investigate these sequences.

Maybe   is a close approximation to the solution. A graph of    is given below.

For illustration purposes, use the approximate solution    and use it so solve  , and plot the result.

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