solving ODE with integration factors

I once again have a disagreement with my textbook (Giggle)

If anyone can find a mistake I would be grateful.

edit: I just realized I probably need to have the integration factor in absolute value and k > 0, but that will propagate through my solution and not effect the work I have done so far.

;

integration factor where k is non zero.

where

This is about where I stopped, I tried expanding the numerators and factoring to reduce my highest power of x by one which worked, but I still have x's in the denominators of my solution which the book does not have, so I guess I have made a mistake already.

The final solution in the book is

Thanks for any help.