Euler-Lagrange - constrained optimization with no solution satisfying constraint?

Question

The questions asks to find the minimum value of the functional $$ \mathcal I [y] = \int_0^1 y'^2 + x^2 \; \text d x $$ subject to the constraint $$ \mathcal J [y] = \int_0^1 y^2 \; \text d x = 2 $$ and boundary conditions $ y(0) = y(1) = 0 $. I started by defining the Lagrange function $ \hat{F} = y'^2+x^2 - \lambda y^2 $ and applying Euler-Lagrange to this to obtain the second order ODE $ y'' - \lambda y = 0 $. This is easily solved by the standard technique of an exponential ansatz having the form $ y = e^{\mu x} $, which gives the solution $ \mu = \pm \sqrt{\lambda} $ and therefore $ y $ must have the form $y = A \sinh (x \sqrt \lambda) + B \cosh (x \sqrt \lambda) $. At this point, I run into problems because when I attempt to substitute the boundary condition back in, I obtain $A = B = 0$ which implies y is identically zero over the domain and then cannot satisfy the given constraint.

Answer 1

With $F = y'^2 + x^2 - \lambda y^2$, the Euler-Lagrange equation should be $y'' + \lambda y = 0$. This gives solutions $y = A \sin(n \pi x)$ for integer $|n|>0$, and certain $A$ satisfying the constraint.