I'm trying to find a solution to a Fredholm equation of the second kind of the form $$f\left(x\right)=g\left(x\right)+\lambda\intop_{a}^{b}\mathcal{K}\left(\frac{t}{x}\right)f\left(t\right)\mathrm{d}t.$$ Specificaly, I'm trying to find a solution for the case of $$\begin{align*} g\left(x\right) & =\frac{1}{x};\\ \mathcal{K}\left(x\right) & =\left(1+x\right)\left[\frac{1+x^{2}}{\left(1+x\right)^{2}}K\left(\frac{2\sqrt{x}}{1+x}\right)-E\left(\frac{2\sqrt{x}}{1+x}\right)\right], \end{align*}$$ where the functions $K(x)$ and $E(x)$ are the complete elliptic integrals of the first and second kinds. Is there any known method for solving these kind of equations? I did not find any in textbooks.
I was thinking we can use the change of variables of $$\begin{align*} t & =e^{u},\\ x & =e^{v}, \end{align*}$$ in order to get the equation to have a new kernel $\tilde{\mathcal{K}}$ of the form $$\tilde{\mathcal{K}}\left(u-v\right)=\mathcal{K}\left(e^{u-v}\right),$$ and then use the transnational invariance of $\tilde{\mathcal{K}}\left(u-v\right)$ to solve using Fourier transform methods for the convolution integral we get. The problem is that the convolution integral has finite bounds, and therefore straightforward Fourier transform does not help.
Is there any way to say something about the solution to the equation? Even approximations might be helpful. I also do not need a solution for $f(x)$ for every $x$, finding $f(a)$ is good enough.
Thank you!