= Solution
A homogeneous linear differential operator satisfies $L(e^\varepsilon u)=e^\varepsilon L(u)$, so the scaling $x\mapsto x$, $u\mapsto e^\varepsilon u$ preserves every homogeneous linear equation. Its infinitesimal generator is \b[$V=u\partial_u$].
On a patch with $u\ne0$, take $s=\log|u|$, $t=x$. Then $V(s)=1$, $V(t)=0$. Set $w=ds/dt=u'/u$. Since $u''/u=w'+w^2$, division by $u$ reduces the second-order equation to the <Riccati equation>
$$
\boxed{\frac{dw}{dt}+w^2+p(t)w+q(t)=0.}
$$
Recover a nonzero solution by $u=C\exp(\int w\,dt)$, 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 $s'$ but not on $s$ itself.
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