Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 29D / c / Solution

A homogeneous linear differential operator satisfies $L(e^{\epsilon }u)=e^{\epsilon }L(u)$, so the scaling $x\mapsto x$, $u\mapsto e^{\epsilon }u$ preserves every homogeneous linear equation. Its infinitesimal generator is $V=u{\partial }_u$.

On a patch with $u\ne 0$, take $s=\log |u|$, $t=x$. Then $V(s)=1$, $V(t)=0$. Set $w=ds/dt=u^{\prime }/u$. Since $u^{\prime \prime }/u=w^{\prime }+w^2$, division by $u$ reduces the second-order equation to the Riccati equationRecover a nonzero solution by $u=C\exp (\int wdt)$, allowing the sign in $C$. The zero solution is separate, and intervals containing zeros of other solutions require separate coordinate patches. The order reduction occurs because the transformed equation depends on $s^{\prime }$ but not on $s$ itself.