Solving a boundary value problem via Euler–Lagrange

The extremal of the functional $I(y(x))=\int_0^1\dfrac{\dot y^2}{x^3}\,dx$, when $x(0)=0$, $x(1)=1$ is: (the boundary conditions are printed with $x$ and $y$ swapped — read as $y(0)=0, y(1)=1$, matching the functional's own variable)