I'm having trouble solving an integral equation. It appears to me to be a homogeneous Fredholm equation of the second kind. However, I'm being told that this can't be a Fredholm equation, because it is non-linear. Could someone help me in trying to figure out how to classify an integral equation as linear or non-linear. Also, I'll post the equation I need to solve below, and it would be great if anyone could also give me some tips on how to try and solve it. Thank you to all who reply.
The equation is
$$\phi(x) = (x^2 - x)\int_0^1 \mathrm{d}y \frac{\phi(y)}{(y-x)^2}$$
Also, is this by chance related to an eigenvalue problem? I know that might sound like a strange question, but I've seen some people treating these as eigenvalue equations.
By the way, I want to solve the equation for $\phi(x)$
This is homogenous Fredholm integral equation of the second kind. It certainly is linear in the function $\phi$, as already observed by Fabian (see comments).
An important observation is that the kernel has a singularity in the integration domain, for $0 \le x \le 1$, which makes the equation a singular Fredholm integral equation of the second kind.
I don't know how to solve your equation, but here is a reference that has a chapter dedicated to singular Fredholm integral equations of both the first and second kind:
See chapter 9, "Some singular integral equations".
I'm not quite sure, but I think they don't have examples with a quadratically diverget kernel, but it may be useful to read that chapter nevertheless.