Examples help but I mostly just want to know what the criteria is for an equation to give non-sinusoidal periodic functions as solutions. https://en.wikipedia.org/wiki/Periodic_function the first image here is non-sinusoidal and it looks like it's not symmetric for the part it repeats.
An example of a sinusoidal solution to an equation is: finding the solution to $\frac{d^2x}{dt^2} = -\frac{kx}{m}$ (SHM).
I can't think of anything that would give such solutions. (I guess I should also ask, are there methods of obtaining such solutions?)
While there are very few differential equations we know how to solve, whenever the second derivative of a variable $\left(\dfrac{d^2x}{dt^2}\right)$ is proportional to the inverse of the variable itself $(x)$ we guess that the solution to the differential equation is most likely sinusoidal. Why?
Well say we have $x(t)=\sin(kt+\phi)$. Then, $x'(t)=k\cos(kt+\phi)$ and $x''(t)=-k^2\sin(kt+\phi)$. Thus, $x''(t)=-k^2x(t)$ and we have that the second derivative of the function is proportional to the original function itself.
Very similarly to the Hooke's Law case, the wave equation has a sinusoidal solution to the differential equation.