Euler Lagrange, weird Second Order DE I have found the Euler-Langrange equation of the functional:
$$\int_{a}^{b} y'^2+y^4\,\mathrm dx$$
to be $y'' = 2y^3$
How do I solve this non-linear DE?

$$y'' = 2y^3$$
Substitute 
$$p=\frac {\mathrm dy}{\mathrm dx} \implies y''=\frac {\mathrm dp}{\mathrm dx}=\frac {\mathrm dp}{\mathrm dy}\frac {\mathrm dy}{\mathrm dx}=p'p$$
Then the equation is separable
$$pp'=2y^3$$
Integrate
$$\int p\,\mathrm dp=2\int y^3\,\mathrm dy$$
$$p^2+K=\frac {y^4}2$$
$$\frac {\mathrm dy}{\mathrm dx}=\pm \sqrt {\frac {y^4}2+C}$$
$$.....$$
