4.17 Variational calculus

Problem: Find the eigenvalues and the eigenfunctions of the functional: \[ \begin{array}{l} \mathcal{T}[y]=\int_{0}^{1}\left(y^{2}+\left(y^{\prime}\right)^{2}\right) d x, \quad y(0)=y(1)=0, \\ \int_{0}^{1} y^{2} d x=0 . \end{array} \]