Solve the following Fredholm integral equation with symmetric kernel : $$y(x)=\lambda\int_0^1 tK(x,t)y(t)dt$$ where $$K(x,t)=\begin{cases} \frac{x}{2t}(1-t^2) &\text{if} \ \ \ \ \ 0\leq x<t \\ \frac{t}{2x}(1-x^2) &\text{if} \ \ \ \ \ t<x\leq 1 \end{cases} $$
I have no idea about how to proceed on this one. In the theory of symmetric kernels, they state that the solution of the equation $$f(x)=\int_a^b K(x,t)y(t)dt$$ where $K(x,t)$ is symmetric and $f$ is known, is given by $$y(x)=\lim_{m\to \infty} \sum_{n=1}^m \lambda_nf_n\phi_n(x)$$ where $\phi_n$ are associated eigenfunctions of the equation and its corresponding eigenvalues are $\lambda_n$ and $\displaystyle f_n=\int_a^b f(t)\phi_n(t)dt$. But here in the problem there is no such known function $f(x)$. So no formal method of solution I'm getting. How to solve this equation at hand? Any help is appreiated.