4.21 Variational calculus

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} J(x(\cdot))=\int_{0}^{T_{0}}\left(\dot{x}^{2}-x\right) d t \rightarrow \text { extr }, \quad x(0)=0, \\ x\left(T_{0}\right)=\xi . \end{array} \]