Transform the following differential equation using new coordinates $y = x$ and $s = \log(t)$ $$2t^2\ddot x + 3t\dot x + xe^{-t} = 2/t $$
Now I understand $\tau(s) = t$ and $s = \sigma(t)$ as well as $\eta(t,x) = y$
but I have only performed a coordinate transformation for first order equations. How is this extended to 2nd order?
I am looking simply for the transformed function, not the solution
I assume you start with taking the second partial derivatives of the coordinates transformations, but any clarity would be great!
Putting $y(s)=x(e^s)$ you have $ x(t)=y (\log t)$, so
$$\dot x(t)=\frac{ \dot y (\log t)}{t}$$
$$\ddot x(t) = \frac{ \ddot y (\log t) - \dot y (\log t)}{t^2}$$
Plug those expressions in your ODE and see what you come up with.