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

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.

• It looks like you solved for the constants incorrectly - there certainly are non-zero solutions to $y'' - \lambda y = 0$ with $y(0)=y(1) = 1$. – B. Mehta Mar 17 '18 at 19:27
• Apologies that was a typo, should have been a $0$. I'll edit it in now – backstrapp Mar 17 '18 at 19:30

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.