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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 $y''=f(y)$?

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    I'm more used to calling this sort of DE's *autonomous*, but anyway..,2011-08-30

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