Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 3 / b / i / Solution

For $x>0$, differentiation of $u=y/x$ givesThus the radial Laplacian equation reduces toBoundedness of $u$ at zero implies $y(0)=0$. Conversely, a regular reduced solution with $y(0)=0$ gives finite $u(0)=y^{\prime }(0)$.

There is an important qualification to the reduction in the printed question: bounded $u$ at infinity requires $y=O(x)$, not bounded $y$. The example in part (ii) actually has $y\sim -x$, so it cannot be the bounded solution described in part (a). The differential equation and origin condition reduce as asserted, and the same Green's function can still be used with the correct growth condition. This is the radial reduction and growth at infinity distinction.