Given some known functions $f(x)$ and $g(x)$, is it possible to find all solutions $\psi(x)$ that satisfy the following constraints?
$$1=\int_{-\infty}^\infty dx\,f(x)\psi(x)$$ $$1=\int_{-\infty}^\infty dx\,g(x)\psi(x)$$
If so, how exactly can this be done?