# Finding particular solution with fourier series

• naggy
In summary, a particular solution in Fourier series is a solution to a differential equation that satisfies a set of initial conditions and is used to find the complete solution. To find a particular solution, one must first find the homogeneous solution and then use a method such as undetermined coefficients or variation of parameters. A particular solution can be used to find the general solution by combining it with the homogeneous solution. The homogeneous solution is a general solution that does not depend on specific initial conditions, while a particular solution is a solution that satisfies the initial conditions. Fourier series can only be used for linear differential equations with constant coefficients and is not applicable for nonlinear or variable coefficient differential equations.
naggy
I was wondering what to do if I have a DE like this

$$x'' + \omega^2x = \cos^2(t) \sin^2(t)$$

I have to decide for what omega it has a solution with period 2pi.

Now to solve this I have to find the Fourier series representation of the right hand side, but the problem is that I get all An = Bn = 0 ??

wait! The right hand side already is a Fourier series

## 1. What is a particular solution in Fourier series?

A particular solution in Fourier series is a solution to a differential equation that satisfies a set of initial conditions. It is used to find the complete solution to the differential equation by combining it with the homogeneous solution.

## 2. How do you find a particular solution using Fourier series?

To find a particular solution using Fourier series, you first need to find the homogeneous solution of the differential equation. Then, you can use the method of undetermined coefficients or variation of parameters to find a particular solution that satisfies the initial conditions.

## 3. Can a particular solution be used to find the general solution in Fourier series?

Yes, a particular solution can be used to find the general solution in Fourier series. By combining the particular solution with the homogeneous solution, you can obtain the complete solution that satisfies the initial conditions.

## 4. What is the difference between a homogeneous solution and a particular solution in Fourier series?

The homogeneous solution is a solution to the differential equation that does not depend on any specific initial conditions. It is the general solution to the homogeneous differential equation. On the other hand, a particular solution is a solution that satisfies a set of initial conditions and is used to find the complete solution to the differential equation.

## 5. Can Fourier series be used for all types of differential equations?

No, Fourier series can only be used for linear differential equations with constant coefficients. It is not applicable for nonlinear or variable coefficient differential equations.

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