4.4 Variational calculus

Problem: Find admissible extremals in the problem with free endpoints:

\begin{array}{l} F (x) = \int_{0}^{1} ({\dot{x}}^{2} + 4 x^{2}) d t \to extr, \\ x (1) = \frac{e^{2} + e^{- 2}}{2} . \end{array}