4.19 Variational calculus

Problem: For the simplest problem of the classical calculus of variations describe the set of admissible functions and investigate the functional for an extremum. \[ \begin{array}{l} \mathrm{J}=\int_{1}^{2}\left(\dot{x}^{2}-t^{2} x\right) d t \rightarrow \text { extr }, \\ x(0)=0, \quad x(1)=0 . \end{array} \]