4.12 Variational calculus

Problem: Find the extremals of the functional: \[ \begin{array}{l} I[y(x)]=\int_{-1}^{1} \sqrt{y\left(1+y^{2}\right)} d x, \\ y(0)=\frac{1}{\sqrt{2}}, \quad y(1)=\frac{1}{\sqrt{2}} . \end{array} \]