For example define: $K=-i\frac{d}{dx}$ (non-discrete spectrum), so:
$$Kf(x)=-i\frac{df}{dx}=kf(x)$$
Define $g(x,k)=kf(x)$, so:
$$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$
$$\frac{\partial{g}}{\partial{f}}f(x)=g(x,k)$$
Which implies (Not sure this helps):
$$Kf(x)=\frac{\partial{g}}{\partial{f}}g(x,k)$$
My question is: How/Can I transform the above (or something else related, maybe using functionals of $f$) to ODE/s, and then solve for the eigenfunctions without use of the eigenvalues?