# Find the periodic solutions of the following differential equations:

I am not sure how to continue with the following question:

Find the periodic solutions of the differential equations (a)$\frac{dy}{dx}+ky=f(x)$, (b) $\frac{d^3y}{d^3x}+ky=f(x)$
where k is a constant and f(x) is a $2\pi$-periodic function.

Consider a Fourier series expansion for $f(x)$ using the complex form, $f(x)=\sum\limits_{n=-\infty}^{n=+\infty} f_ne^{inx}$
and try a solution of the form: $y(x)=\sum\limits_{n=-\infty}^{n=+\infty} y_ne^{inx}$

So far, I've taken the first, second and third derivative of $y(x)$, and tried to substitute these into the given differential equations. Since the summation is from $-\infty$ to $\infty$, I figured no rewriting of the summation terms is needed. Therefore, $y'(x)$ and $y''(x)$ and so on would be easier to write down. Now for the solution, what I get for a) is that
$$iny_n+ky_n=f_n$$

From this point on, I'm not sure how to continue, but I think you have to solve the equation for $y(x)$ given the fact that $f(x)$ is periodic. I know how to find the solutions of differential equations, but for this periodic one, I am not sure what I should find and how I should do it.

Perhaps you can help. I only need a general way of solving these kinds of problems, and maybe a tiny example to illustrate how it's done. Anyway, I'd appreciate it if at least someone could help me with question a). I'd be able to do b) that way.

If $y(x) =\sum\limits_{n=-\infty}^{n=+\infty} y_ne^{inx}$, then $y' =\sum\limits_{n=-\infty}^{n=+\infty} iny_ne^{inx}$, $y'' =\sum\limits_{n=-\infty}^{n=+\infty} -n^2y_ne^{inx}$, and $y''' =\sum\limits_{n=-\infty}^{n=+\infty} -in^3y_ne^{inx}$.
For the second equation, $f_n = -in^3y_n+y_n =y_n(1-in^3)$ so $y_n =\dfrac{f_n}{1-in^3} =\dfrac{f_n(1+in^3)}{1+n^6}$.
• One could use the integral expressions for the Fourier coefficients $f_n$ and (potentially) resum $y=\sum_n \frac{f_n e^{i n x}}{1-i n^3}$ in the cubic case to obtain $y(x)$ as an integral transform of $f(x)$. – Semiclassical May 29 '17 at 21:47