I've been investigating a lot of interesting physically inspired second order differential equations that don't have t in the equation at all and thus are separable (one example being $y''=-\frac{k}{y^2}$).
I know that if y' = f(y) I can solve this by $\int {\frac {dy}{f(y)}} = \int dt$
For the second order equation y'' = f(y) I have been removing a variable since t is not involved by the following steps, but it can get messy.
y' = v v' = f(y) \frac{v'}{y'} = \frac{\frac{dv}{dt}}{\frac{dy}{dt}} = \frac{dv}{dy} = \frac{f(y)}{v}
$\int{v dv} = \int{f(y) dy}$
And after solving for v(y), then trying to solve for y(t), by substitution and a further 1st order separable equation.
I was wondering if there is some easier approach for a 2nd order of the form y''=f(y)?