4.8 Variational calculus

Problem: Let \( M=C[-1 ; 1] \) - the class of the functions \( y(x) \), continuous on segment \( [1 ; 1] \). Given functional \( I[y(x)]=\int_{-1}^{1} \varphi(x, y) d x \), where \( \varphi(x, y) \) - is the function, defined and continuous for all \( x \in[-1 ; 1] \) and real \( y \). For the given function \( \varphi(x, y) \) select couple of functions \( y(x) \) and find the corresponding values of the functional. \[ \varphi=\frac{x}{2-y^{2}} \text {. } \]