4.23 Variational calculus

Problem: Solve the simplest problem of the classical calculus of variations: \[ \begin{array}{l} \int_{0}^{T_{0}}\left(2 \dot{x}^{2}+x^{2}-4 x \sin t\right) d t \rightarrow \text { extr }, \\ x(0)=0, \quad x\left(T_{0}\right)=\xi . \end{array} \]