If we introduce this dummy variable t and show that if we have after manipulating we're almost done showing that its homogeneous. The last part is to show that we get after manipulating AND that the we have the same exponents. If the exponents are different, OR we can't show that or , then its not homogeneous.
Does this help?
I think i'm understanding it, but there's just something else that's puzzling.
and we're comparing this with:
Comparing coefficients we get:
Is this equal to:
For the next part:
and since the -2 on the LHS does not contain a t then there is no value of for which this is true.
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.
Is that about right?