I think i'm understanding it, but there's just something else that's puzzling.

We have:

and we're comparing this with:

Comparing coefficients we get:

and

Is this equal to:

and

?

For the next part:

This gives:

and since the -2 on the LHS does not contain a t then there is no value of

for which this is true.

**Correct**
I haven't tried it with the y's yet since it has to satisfy both these conditions to be homogeneous. Since this one doesn't work the DE is non-homogeneous.

**Correct. You will end up with the same situation with the y's, where there is no value where this would be true.**

Is that about right?