I'm not entirely sure I understand your question. However, I would note that
The DE becomes
We can now integrate directly, once, to obtain
or re-defining we have
Can you continue?
Hmm. That might be difficult. The substitution yields the DE
which is technically in the form you're looking for, with and
The substitution gives you the same DE, actually, just one order lower:
In this case, you have and
I don't know if that helps you or not. That's a strange form to try to get the DE into - at least I'm not familiar with it. It looks semi-Ricatti, but in a second-order form. I don't know.
Simply dividing through by yields the immediately integrable expression
Other than this, I'm about out of ideas.