Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 3 / 11A / ii / Solution

The radial Poisson equation with constant source one has $\psi =r^2/6+a+b/r$. The boundary conditions giveSubtracting gives $b=1$, then $a=-1/6$. HenceIts radial Laplacian is one and both endpoint values are one. The prescribed values ensure continuity with the adjacent region solutions, but do not impose derivative continuity.