Solve of the differential equation $y'=-\frac{x}{y}+\frac{y}{x}+1$ I've tried to solve this equation, and in the course of solving any problems. Please help me understand.
$$y'=-\frac{x}{y}+\frac{y}{x}+1$$
Results in a normal form.
$$y'=-\frac{1}{\frac{y}{x}}+\frac{y}{x}+1$$
Make replacement.
$$\frac{y}{x}=U$$
$$y'=U'x+U$$
Substitute into the original equation.
$$U'x+U=-\frac{1}{U}+U+1$$
$$\frac{dU}{dx} \cdot x = -\frac{1}{U}+1$$
What to do with it?

Expanding on Sonnhard's hint, on the left hand side you have $$\Big(1+\frac{1}{u-1}\Big)du$$
Integrate this gives you $u+\text{ln}|u-1|+C$, where $C$ is some arbitrary constant. 
