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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.

am_11235...
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  • This does not satisfy the uniqueness condition! – Math-fun Jan 13 '20 at 14:44
  • What uniqueness condition? By the way, I have seen that reducing the BVP $x^2y''+xy'+(\lambda x^2-1)y=0$ with $y(0)=y(1)=0$ by Green's function technique gives us the above integral equation. So solution will be modified Bessel of order $1$. Solution must exist. I am just unable to find this solution by integral equation technique, that's all. – am_11235... Jan 13 '20 at 14:51

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