$\displaystyle \frac{d^2 x}{dt^2}+4\frac{dx}{dt}+9x=2cos3t$

Is this ordinary, linear differential equation homogeneous? I was thinking about this because 2cos3t can =0 at $\displaystyle \frac {\pi}{2}+n\pi$ so at these points could it be considered homogeneous?

$\displaystyle \frac{dy}{dx}=\frac{-y(2-3x)}{x(1-3y)}$

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 $\displaystyle \frac{dy}{dx}=F(\frac{y}{x})$". $\displaystyle \frac{y}{x}$ is present on the RHS, however $\displaystyle \frac{2-3x}{1-3y}$ cannot be written in this way. Therefore the equation is non-homogeneous.

Alternatively this can be rearranged to $\displaystyle \frac{dy}{dx}+\frac{y(2-3x)}{x(1-3y)}=0$. 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 $\displaystyle \frac{\delta u}{\delta t}=\frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}$ autonomous?

(i'm not sure if $\displaystyle \delta $ 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 $\displaystyle \frac {dy}{dx}=(y) $. 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?