I was recently playing around with a problem for fun(as one does in grad school) and came across an nonhomogeneous nonlinear ODE and I'm hoping someone can explain how to solve it: $$ y^{2}-\left(1+f(x)\right)y\frac{dy}{dx}+f(x)\frac{dy}{dx}^{2} = \left( y-\frac{dy}{dx} \right) \left( y-f(x)\frac{dy}{dx} \right)=k $$ I've looked through most of my books that I would think touch on it... 2 different undergrad ODE books(Zill and Haberman), 2 Dynamical Systems books (Verhulst and Sternberg), and Nonlinear ODEs by Jordan and Smith... but with no luck.
I'm really at a loss on this. I just don't know how to go about solving an ODE of this form. Nor, it appears, do my textbooks.
For w/e it's worth, thus far, Dr. Google has only delivered answers for how to solve cases without a power on the derivative. Which is what the profs in my dept, whom I've asked thus far, have also scratched their heads at.
Any help would be greatly appreciated.
Anybody? Any suggestions?
I'm pretty much out of ideas other than plugging in different functions to see what happens.