Derive the Euler-Lagrange equation for the integral

$$\int_{x_{0}}^{x_{1}} f\left(x, u, u^{\prime}\right) d x$$

where $u\left(x_{0}\right)$ is allowed to float, $\partial f /\left.\partial u^{\prime}\right|_{x_{0}}=0$ and $u\left(x_{1}\right)$ takes a given value.

Given that $y(0)$ is finite, $y(1)=1$ and $y^{\prime}(1)=1$, find the stationary value of

$$J=\int_{0}^{1}\left(x^{4}\left(y^{\prime \prime}\right)^{2}+4 x^{2}\left(y^{\prime}\right)^{2}\right) d x$$