Solving a separable 2nd order differential equation (can a similar technique be used)?

Solution 1:

There is no general approach for all ODE's of the form $ y'' = f(y) $. In fact, even when the functions $f(y)$ are restricted to relatively simple functions, the solutions can be very exotic. For example, if $f(y) = 2k^3 y^3-(1+k^2)y$ where $k$ is some fixed constant, the resulting solution (with initial conditions $y(0)=y'(0)=0$ ) is something called the Jacobi elliptic function $\operatorname{sn}(z,k)$. That's just one case for what is a relatively simple $f(y)$ (it was a 3rd degree polynomial). People devote a lot of effort into studying methods to solve DE's like yours, for specific special families of functions $f(y)$, with each special family having special techniques to solve them.