A function $y(x)$ is chosen to make the integral

$$I=\int_{a}^{b} f\left(x, y, y^{\prime}, y^{\prime \prime}\right) d x$$

stationary, subject to given values of $y(a), y^{\prime}(a), y(b)$ and $y^{\prime}(b)$. Derive an analogue of the Euler-Lagrange equation for $y(x)$.

Solve this equation for the case where

$$f=x^{4} y^{\prime \prime 2}+4 y^{2} y^{\prime}$$

in the interval $[0,1]$ and

$$x^{2} y(x) \rightarrow 0, \quad x y(x) \rightarrow 1$$

as $x \rightarrow 0$, whilst

$$y(1)=2, \quad y^{\prime}(1)=0$$