[SECOND EDIT]: See next post for a correction in the OP.
Hmm. Strange. I don't get quite the same integration you do. Using your substitution, I get
The integral on the left is intractable. Have I made a mistake somewhere?
I wonder if the opposite substitution, that of would give you a more do-able integral. *Working it out...*
[EDIT]: Nope, that integral is just as intractable.
It looks to me like the best you can do is reduce the DE to quadratures.