[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.