Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 39C / c / Solution

The simultaneous diagonalization from part (a) applies to both $B$ and $C$. With ${\theta }_k=k\pi h$,andMultiplying the $j$th block equation by $Q^{-1}$ and writing $v_j=Q^{-1}{u}_j$ and $c_j=Q^{-1}{b}_j$ giveswith the absent boundary terms omitted. This is exactly system (3).