Past exam of the short options course of the University of Oxford / 2017 / 6 / a

Apply a constant Lagrange multiplier to the isoperimetric constraint, using the augmented functional $J-\lambda (I-1)$. Its integrand isThe Euler-Lagrange equation gives $-2qy-2\lambda wy-2(p{y}^{\prime }{)}^{\prime }=0$, henceThe multiplier rule applies because the constraint derivative in direction $y$ is $D{I}_y[y]=2I[y]=2$, so the constraint derivative is not zero. The prescribed endpoint values remove boundary variations.

Multiply the Sturm-Liouville equation by $y$ and integrate. The endpoint values and integration by parts giveSince $I[y]=1$,Thus the multiplier equals the stationary value of $J$ at the extremal, as in stationary values of a quadratic functional with a quadratic constraint. It is also the corresponding Rayleigh quotient.

Multiply the equation for $y_n$ by $y_m$ and that for $y_m$ by $y_n$, then subtract. Integrating the resulting identity yieldsAll endpoint terms vanish because both functions satisfy the zero Dirichlet boundary conditions. Since the multipliers differ,This is the weighted orthogonality of Sturm-Liouville eigenfunctions.