Integral equation $\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$

Question

In Sewall Wright's Evolution of Mendalian Population, the equation for the nonrecurrent mutation is $$\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$$ where $n>0$ and $\binom a b$ is the binomial coefficient. We are to solve for a nonzero function $\phi\ne0$. It is an eigenvalue problem of a compact operator. But what is a general method for solving this integral equation? dirichlet Is there a transform, say Mellin transform, that does the trick?