Homogeneous Differential Equations.

Is this ordinary, linear differential equation homogeneous? I was thinking about this because 2cos3t can =0 at so at these points could it be considered homogeneous?

This is an ordinary, non-linear differential equation. However, is it homogeneous?

My book defines a homogenous equation as "a differential equation that can be written as ". is present on the RHS, however cannot be written in this way. Therefore the equation is non-homogeneous.

Alternatively this can be rearranged to . Since this is non-linear then although it equals 0, it is not homogeneous.

I just wanted to see if I was thinking of this the correct way.

Your input would be very much appreciated!

P.S: Sorry to just add this in under this heading but I got stuck on this as well:

Is autonomous?

(i'm not sure if is the right symbol for a partial DE, but it looked the most like the one i've written down).

Once again, consulting the almighty book of knowledge I get that . However it doesn't mention anything regarding partial DEs! I gather that if the equation does not contain any t, x and y's then it is autonomous, however it is difficult to see if the equation actually does.

When push comes to shove, I would say this is autonomous. Is this correct?