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?

HINT: we have
$$\frac{du}{1-\frac{1}{u}}=\frac{dx}{x}$$
the term $$\frac{u}{u-1}=\frac{u-1+1}{u-1}=1+\frac{1}{u-1}$$
