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?

$$y''=2y^3$$
$$2y''y'=4y^3y'$$
$$y^4=y'^2+c_1$$
$$y'=\pm \sqrt{y^4-c_1}$$
$$x=\pm\int \frac{\mathrm dy}{\sqrt{y^4-c_1}}+c_2$$
This is an elliptic integral : http://mathworld.wolfram.com/EllipticIntegral.html
The inverse function $y(x)$ involves the sn Jacobi elliptic function :
http://mathworld.wolfram.com/JacobiEllipticFunctions.html
$$y(x)= C_1 \text{sn}\left(C_1(x+c_2)\:|\:-1 \right)$$
with $C_1=c_1^{1/4}$
