By the chain rule,

which follows from
.
is then a function of
; denote this function by
. Then by the product rule,
![(\mathrm d^2y)/(\mathrm dx^2)=(\mathrm d)/(\mathrm dx)\left[\frac1x(\mathrm dy)/(\mathrm dt)\right]=-\frac1{x^2}(\mathrm dy)/(\mathrm dt)+\frac1x(\mathrm df)/(\mathrm dx)](https://img.qammunity.org/2020/formulas/mathematics/college/aeai64wlpljiakcj99kwvn0929civ09dej.png)
and by the chain rule,

so that

Then the ODE in terms of
is

The characteristic equation

has two roots at
and
, so the characteristic solution is

Solving in terms of
gives
