4.13 Variational calculus

Problem: Check the validity of the Jacobi condition: \[ \begin{array}{l} I[y(x)]=\int_{0}^{a}\left(y^{2}-y^{2}\right) d x, \quad a>0, \\ a \neq \pi k, \quad y(0)=0, \quad y(a)=0 \end{array} \]