A fourth-order Euler–Poisson equation solved exactly by $\sin x$

The curve which extremizes the functional $I[y(x)]=\int_0^{\pi/4}(y"^2-y^2+x^2)dx$ under the conditions $y(0)=0, y'(0)=1, y(\pi/4)=y'(\pi/4)=1/\sqrt2$ is: