I have been struggling with the following problem I came across in a textbook. I believe that it is necessary to use the Euler-Lagrange Equation. Any help would be greatly appreciated.
Let $F$ be the functional defined by $$F(u) = \int_0^\pi u'(t)^2 - c u(t)^2\, dt.$$ Assume that $u(0) = 0 = u(\pi)$. Find a candidate for the weak minimum of $F$ if we let $c$ be a perfect square.