Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 20 / c

For $u=XY$, the separated equations areOn a region containing the prescribed points, take $0<x<\pi$ and $0<y<\pi /2$. Integration givesThe two specified values, verified in the original PDF, giveDividing these relations eliminates $C$:The right-hand side is the square of the positive base on the left, which differs from one. Thus $\lambda =2$, and either original condition gives $C=1$. ThereforeIts partial derivatives satisfy $u_x/\sin x=2u$ and $\tan y{u}_y=2u$, verifying the equation wherever its coefficients are defined. Substitution at the two assigned points gives $1/(4e^{\sqrt{3}})$ and $3/(4e)$ exactly. The formula extends to any region where $\sin x\ne 0$ and $\cos y\ne 0$.