Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 13E / d / Solution

Take $n=1$ in part (c). The original equation $w_{xx}+w=0$ has the solution $\sin x=\sqrt{1-z^2}$. The solution $w_2$ is normalized by $w_2/(1-z{)}^{1/2}\to 1$ as $z\to 1$, soComparing this with the formula for $w_2$ and using $u=(1-z)/2$ givesThis also agrees with the general identity $F(A,C;C;u)=(1-u{)}^{-A}$.