Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 1 / ii / c / Solution

Expand the forcing and the solution in the normalized eigenfunctions: $1={\sum }_n{b}_n{y}_n$ and $y={\sum }_n{c}_n{y}_n$. Their coefficients areSince all ${\lambda }_n=n^2{\pi }^2-1$ are nonzero, the equation gives $c_n=b_n/{\lambda }_n$. Hence the required Sturm-Liouville eigenfunction expansion isThis is constant forcing in a logarithmic Dirichlet Sturm-Liouville problem. The coefficients are $O(n^{-3})$, so the solution series converges uniformly and has zero endpoint values. Applying $L$ is justified in the weighted square-integrable sense; the forcing sine series equals one at interior points, while its endpoint sums are zero. The equation is imposed in the interior, so this does not contradict the boundary conditions.

As an independent check, solving $-Y^{\prime \prime }-Y=1$ directly givesIt has both zero endpoint values and satisfies $Ly=1$. The invertibility of the Dirichlet operator makes it the same solution as the expansion.