4.2 Variational calculus

Problem: Find admissible extremals of the functional: \[ \begin{array}{l} F(x)=\int_{-1}^{1}\left(\dot{x}^{2}+x^{2}-2 t x\right) d t \rightarrow \text { extr }, \\ x(-1)=-1, \quad x(1)=1 . \end{array} \]