There are a few posts on this forum that deals with the transformation of non-autonomous differential equations into autonomous differential equations, in particular: Transforming differential equations into autonomous first-order systems and When can a non-autonomous system NOT be re-written as an autonomous system?.
Mainly I'm currently tasked to transform the Bessel Equation$$t^2u^{''}(t) + tu^{'}(t) + (t^2 - \alpha^2)u(t) = 0$$into a system of autonomous differential equations of first order. The transform to first order is quite simple by defining $z_1(t) = u(t)$ and $z_2(t) = u'(t), yielding\begin{align*}z_1^{'}(t) &= z_2(t) \\end{align*}\begin{align*}z_2(t) &= -\frac{1}{t}z_2(t) - (1 - \frac{\alpha^2}{t^2})z_1(t) \\end{align*}
To make this system autonomous, we can introduce some parametrization $s(t)$ with $s^{'}(t) = 1$, expanding the above system to\begin{align*}z_1^{'}(s) &= z_2(s) \\\end{align*}\begin{align*}z_2(s) &= -\frac{1}{s}z_2(s) - (1 - \frac{\alpha^2}{s^2})z_1(s) \\\end{align*}\begin{align*}s^{'}(t) = 1\end{align*}However I don't see how this system is in any way easier to analyse or to solve. The equation $z_2(s) = -\frac{1}{s}z_2(s) - (1 - \frac{\alpha^2}{s^2})z_1(s)$ didn't get any easier to tackle with the common solution methods and I'm just kind of stumped why this step is useful (not just for the Bessel Equation but also for the general case).