Derive the Euler-Lagrange equation for the integral

$$I[y]=\int_{x_{0}}^{x_{1}} f\left(y, y^{\prime}, y^{\prime \prime}, x\right) d x$$

when $y(x)$ and $y^{\prime}(x)$ take given values at the fixed endpoints.

Show that the only function $y(x)$ with $y(0)=1, y^{\prime}(0)=2$ and $y(x) \rightarrow 0$ as $x \rightarrow \infty$ for which the integral

$$I[y]=\int_{0}^{\infty}\left(y^{2}+\left(y^{\prime}\right)^{2}+\left(y^{\prime}+y^{\prime \prime}\right)^{2}\right) d x$$

is stationary is $(3 x+1) e^{-x}$.