[SOLVED] First order homogeneous ODE

Another one from G.F.Simmons' "Differential Equations": Chapter 2 Miscellaneous Exercises no. 20:

This can be put in the format:

Substituting , from which

which leads to:

This leads to the messy:

Flogging through the calculus and after tidying up:

which is miles away from the given answer:

(although that looks promising)

If I try solving it by doing:

and dividing T&B by and substuting it's even worse.

I must be doing something fundamentally wrong in the first few lines but for the life of me I can't work out what.