I have the following DE:
$ xy' = y + x\cos^2\left(\frac{y}{x}\right) $
I then rule out the possible methods of solving it:
- Not separable
- Not homogeneous
- Not exact
- Possible integrating factor in $x$? No.
- Possible integrating factor in $y$? No.
- Not linear
Above are the only ways I have learned to solve DEs. To help, I've rewritten the DE in this form:
$ M(x,y)dx + N(x,y)dy = 0 $ $ \left(y + x\cos^2\left(\frac{y}{x}\right)\right)dx - xdy =0 $ $ M_{y} = 1 - 2\cos\left(\frac{y}{x}\right)\sin\left(\frac{y}{x}\right) $ $ N_{x} = -1 $
I'm completely lost now. I can't seem to find any integrating factors (the $\frac{y}{x}$) inside the trigs are making it so that I can't get things only in terms of $x$ or $y$.