Application of
Fourier Series to Differential Equations

Since the beginning Fourier himself was interested to find a powerful tool to
be used in solving differential equations. Therefore, it is of no surprise that
we discuss in this page, the application of Fourier series differential
equations. We will only discuss the equations of the form

y(n)
+ an-1y(n-1) + ........+
a1y' + a0 y =
f(x),

where f(x) is a -periodic function.

Note that we will need the complex form of Fourier series of a periodic
function. Let us define this object first:

Definition. Let f(x) be -periodic. The complex Fourier series of f(x) is

where

We will use the notation

If you wonder about the existence of a relationship between the real Fourier
coefficients and the complex ones, the next theorem answers that worry.

Theoreme. Let f(x) be -periodic. Consider the real Fourier coefficients and of f(x), as well as the complex Fourier coefficients
. We have

The proof is based on Euler's formula for the complex exponential function.

Remark. When f(x) is 2L-periodic, then the
complex Fourier series will be defined as before where

for any .

Example. Let f(x) = x, for and f(x+2) =
f(x). Find its complex Fourier coefficients .
Answer. We have d0 = 0 and

Easy calculations give

Since , we get
. Consequently

Back to our original problem. In order to apply the Fourier technique to
differential equations, we will need to have a result linking the complex
coefficients of a function with its derivative. We have:

Theorem. Let f(x) be 2L-periodic. Assume that
f(x) is differentiable. If

then

Example. Find the periodic solutions of the differential equation

y'
+ 2y = f(x),

where f(x) is a -periodic function.
Answer. Set

Let y be any -periodic solution of the differential equation. Assume

Then, from the differential equation, we get

Hence

Therefore, we have

Answer. Because

we get with

Let y be a periodic solution of the differential equation. If

then . Hence

Therefore, the differential equation has only one periodic solution

The most important result may be stated as:

Theoreme. Consider the differential equation

where f(x) is a 2c-periodic function. Assume

for . Then the differential equations has one 2c-periodic solution
given by

where

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