Variational Calculus Optimization Problem with x-independent Lagrangian Solution via Beltrami Identity

Question

Optimize $$\int_0^1 y^2 (y')^2 dx$$ subject to $y(0)=0, y(1)=1$

For x-independent Lagrangians it is easier to use Beltrami Identity: $$ F-y'{\frac{\partial F}{\partial y'}}=C $$

Answer 1

According to Beltrami's formula

$$ y^2(y')^2-y'(y^2 2 y')= -y^2(y')^2= C $$

then

$$ \left(\frac 12 \frac{d}{dx}y^2\right)^2= C_1 $$

or

$$ \frac 12 \frac{d}{dx}y^2 = C_2 $$

and

$$ \frac 13 y^3=C_3x+C_4 $$

and with the boundary conditions

$$ y = \sqrt[3]{x} $$