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

With $u=XY$, divide the separated equation on $x\ne 0$, $y\ne 0$ to obtainThus $X^{\prime }/X=\lambda x{e}^{-x}$ and $Y^{\prime }/Y=\lambda /y$. Taking antiderivatives,Choose $\lambda =1$ and the signed branch $Y=y$. A nonconstant solution, smooth for every real $x,y$, isIndeed $u_x=x{e}^{-x}u$ and $u_y=\exp [-(x+1)e^{-x}]$, so $e^x{u}_x-xy{u}_y=0$. This verification includes the axes excluded temporarily during separation.