Solve $y'(x)=\frac{y^2(x)/x^2}{1+y(x)/x}$ via substitution

Question

Assignment: Find the solution of the following ODE:

$$y'(x)=\frac{y^2(x)/x^2}{1+y(x)/x}.$$

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I've tried the substitution $RHS=:v(x,y)$, as well as a few others but doesn't seem to work in simplifying the differential equation, since, for example, the former results in $$\frac{dv}{dx}=\frac{2y\frac{dy}{dx}(x+xy)-y^2((x+xy)\frac{dy}{dx})}{x^2(1+y)^2}$$ which doesn't seem helpful.

Answer 1

Let $y=vx$. Then you get $$v+xv'=\frac{v^2}{1+v}.$$I think you can take it from here.

Answer 2

Hint

Set $z(x)=\frac{y(x)}{x}$. Then your equation becomes $$xz'(x)+z(x)=\frac{z(x)^2}{1+z(x)}\iff z'(x)(1+z(x))=-\frac{1}{x}$$