Given $\int_{-\pi/4}^{\pi/4}(y''^2-y^2)dx$ with the boundary conditions:
$y(-\pi/4)=y(\pi/4)=1$, $y'(-\pi/4)=-y'(\pi/4)=1$
Because th function $f(y,y',y'',x)$ involves a second derivative please re derive the Euler equation from scratch, starting by adding the variation $\alpha\eta(x)$.