ODE with change of variable

Question

I'm trying to solve $$y'=\frac{x+y}{y}.$$ I then tried the change of variable $$ v=\frac{y}{x},$$ and -I think- $$y'=xu'+u$$ I went on few lines but then I got struck since the thing was clearly not going in the direction of the solution. What did I do wrong?

Answer 1

hint

The equation is

$$\frac{uu'}{1+u-u^2}=\frac{1}{x}$$

observe that

$$\frac{u}{u^2-u-1}=\frac{a}{(u-u_1)}+\frac{b}{(u-u_2)}$$

with $u_1=\frac{1-\sqrt{5}}{2}$ $$u_2=\frac{1+\sqrt{5}}{2}$$ $$b=\frac{-u_2}{u_1-u_2}=\frac{u_2}{\sqrt{5}}$$ $$a=\frac{-u_1}{\sqrt{5}}$$

After integration, we get

$$(u-u_1)^a(u-u_2)^b=\frac{\lambda}{x}$$ solve for $u$.

Answer 2

With your Substitution we get $$u+u'x=1+\frac{1}{u}$$ this is $$xu'=\frac{1+u-u^2}{u}$$

Can you finish?