4.4 Variational calculus

Problem: Find admissible extremals in the problem with free endpoints: \[ \begin{array}{l} F(x)=\int_{0}^{1}\left(\dot{x}^{2}+4 x^{2}\right) d t \rightarrow \text { extr }, \\ x(1)=\frac{e^{2}+e^{-2}}{2} . \end{array} \]