Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 20 / a / ii / Solution

With the same separation of variables ansatz, $x{X}^{\prime }Y-2yX{Y}^{\prime }=0$ givesOn $x,y>0$, integration yields $X=A{x}^{\lambda }$ and $Y=B{y}^{\lambda /2}$. The choice $\lambda =2$, $AB=1$ gives a polynomial that extends smoothly to all real $x,y$:It is nonconstant and real everywhere, and $x{u}_x-2y{u}_y=2x^{2}y-2x^{2}y=0$, including on the axes.