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

where

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 2*L*-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 *d*_{0} = 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 2*L*-periodic. Assume that *f*(*x*) is differentiable. If

then

**Example.** Find the periodic solutions of the differential equation

where

Let

Then, from the differential equation, we get

Hence

Therefore, we have

**Example.** Find the periodic solutions of the differential equation

we get with

Let

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

for . Then the differential equations has one 2

where

**
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

**Author**: M.A. Khamsi

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