How do you get the characteristic equation of trickier Euler equations?

When dealing with 2nd order ODE's with constant coefficients, you have this:

which you can get the characteristic equation like so:

Now, when dealing with Euler equations, you have this:

Here's where things get dicey.... When you have it in this nice form, you can get the characteristic equation like so:

However, if you have an Euler equation like this:

...now it's a lot nastier. How do you deal with Euler equations like this?

Can you just do a substitution like

...and revert it back at the end?