I am trying to solve an integral equation of the form:
$$A = \int_0^1 \int_0^1 \mathrm{d}x~\mathrm{d}y~\rho(x) \rho (y) xy,$$
where $A$ is a known constant. I am trying to find the unknown function $\rho(x)$, s.t. $\rho(x) \neq $ const. Any tips on how to approach solving this equation would be greatly appreciated.
The right side factors as $$\left(\int_0^1 dx\; \rho(x) x\right)^2$$ Assuming you are working over the reals, you need $A \ge 0$, and then $\rho(x)$ is any function such that $$ \int_0^1 dx\; \rho(x) x = \pm\sqrt{A}$$
Take any function $f$ such that $c = \int_0^1 dx\; f(x) x$ exists and is nonzero, and let $$ \rho(x) = \pm\frac{\sqrt{A}}{c} f(x)$$ If $A \ne 0$ this gives all the solutions.