I was thinking what if I had a differential equation of the form:
$$\frac{d^2u}{dx^2} + vu(x) = 0 $$
where $v(y(x))$, that is $y$ is a function of $x$. What are the possible restrictions that I can put on this differential equation so that it admits a solution? Has anyone come across any differential equations that contain an implicit function?
I'm a bit lost with your notation changes but if you mean $\frac{d^2y}{dx^2}+f(y(x))=0$, then if you multiply by $\frac{dy}{dx}$ you can integrate once (as long as $f$ is nice enough. The hard bit is usually solving $\frac{1}{2}(\frac{dy}{dx})^2 = const + \hat f$, where $\hat f$ is the integral of $f$. E.g. if $f(y)=\cos(y)$ then you get $\frac{1}{2}(\frac{dy}{dx})^2 = const + \sin(y)$.