Solution of a first order non linear differential equation I am trying to find the extrema of the integral below 
$$
I= \int_0^1 y^2 \mathrm dx
$$
under the conditions
$$
\int_0^1 \left(\frac{dy}{dx}\right)^2 \mathrm dx =1
$$
and y(0)=y(1)=0
Using Lagrange multiplier λ, equivalently, I can find the extrema of the below quantity 
$$
I*= \int_0^1 y^2 \mathrm dx+λ\int_0^1 \left(\frac{dy}{dx}\right)^2 \mathrm dx
$$
Since there is not explicit dependence on x, by using beltrami identity
$$
F-y'\frac{\partial F}{\partial y'} = C
$$
the problem above reduced to the solution of the 2nd order differential equation
$$
2λ(y’)^2y’’-λ(y')^{2}- y^{2}-c=0.
$$
And after the substitution u=dy/dx 
The second order differential equation above become the first order one below
$$
2λu^{3}u’-λ(u)^{2}- y^{2}-c=0.
$$
where the differentiation refers to y.
And here is where I got stuck. 
Can someone give me some hint on how I can procced with that equation?
Many thanks in advance!

The Beltrami equation can not contain a second derivative, where should it come from? It should evaluate to
$$
y^2-λy'^2=C
$$
which easily leads to solutions in terms of trigonometric or hyperpolic functions.
