Solving a differential equation using a substitution

Question

So I have:

$$\frac {dy} {dx} = \dfrac x {(x+y)}$$

dy/dx = du/dx * dy/du

y = ux, so dy/du = u*du/dx + x

I'm not really sure where to go from here, can anyone help?

Answer 1

$$y' = x/(x+y)$$ Substitute $y=tx$ and $y'=t'x+t$

$$t'x+t=\frac 1 {t+1}$$ $$t'x=\frac 1 {t+1}-t=\frac {(-t^2-t+1)}{t+1}$$ It's separable $$\int \frac {t+1}{(-t^2-t+1)}dt=\int \frac {dx}{x}=\ln|x|+K$$

Answer 2

Compute $dy/dx$ instead. Then $dy/dx = du/dx+u$ and you have

$$\frac{du}{dx} +u = \frac{1}{1+u}$$

which is easily separable.

Answer 3

Your equation is $$ y'= \frac {x}{x+y}$$ $$ y=ux \implies y'=u+xu'$$

$$u+xu'= \frac {x}{x+ux}=\frac {1}{1+u}$$

$$ \frac {(u+1)du}{u^2+u-1} = \frac {-dx}{x} $$

Separable.......