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, 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:

• David Porter, David Stirling: "Integral Equations. A practical treatment, from spectral theory to applications".

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.

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:

• David Porter, David Stirling: "Integral Equations. A practical treatment, from spectral theory to applications".

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.