I'm trying to find a solution to a differential equation of an unusual form: $$g(x) u_1(x)=\int_a^b K(x,y) u_2(y) dy$$
where $g(x)$ and $K(x,y)$ are known and $u_1(x)$ and $u_2(x)$ are complex exponentials of magnitude 1, i.e. $u_n(r)=e^{\iota \phi_n(r)}$
I don't need an analytical, just a method of solving this using python and numpy/scipy
The actual equation I'm trying to solve is:
$$2 \pi^{\frac{3}{2}}\sqrt{2 \rho(x)}e^{i \phi_1(x)}=\int_{-a}^a\frac{e^{\iota x \cdot y}}{\sqrt{e^{\frac{^{|y|^2}/_{2b}-c}{d}}-1}}\cdot e^{\iota \phi_2(y)} dy$$
with $\phi_1(x)$ and $\phi_2(y)$ as the unknown functions where both must be real valued functions.
I can't find any thing like this from searching google, just the Fredholm equation and Volterra equation.