Clamped fourth-order variational extremal on a quarter period ($J[y]={\int }_0^{\pi /2}(y^{\prime \prime 2}-y^2+x^2)dx$)

With values $y(0)=y(\pi /2)=y^{\prime }(\pi /2)=0$ and $y^{\prime }(0)=1$, the higher-order Euler-Lagrange equation is $y^{(4)}-y=0$. Write $c=\cosh (\pi /2)$ and $s=\sinh (\pi /2)$. Solving the four linear boundary equations gives $y=[c(\cos x-\cosh x)+(1-s)\sin x+(1+s)\sinh x]/2$. The added function $x^2$ does not affect the first variation.