TITLE: Solving a non-linear ODE
QUESTION [0 upvotes]: Find a general solution to the ODE $$\left(y'' - \frac{(y')^2}{y}\right)^2 + (y')^2 = y^2$$

I know the answer to this one, but I don't know a good method to derive it other than simply guessing.  I'm curious what other people can figure out.

REPLY [4 votes]: Let $u = \dfrac{y'}{y}$, then $u' = \dfrac{y''}{y} - \dfrac{(y')^2}{y^2}$. The equation reduces to
$$ (u')^2 + u^2 = 1 $$
which gives 
$$ u(x) = \cos(x + \phi) $$
and
$$ y=A\exp\big(\sin(x+\phi)\big) $$
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