Homogeneous DE simplification

This is a basic simplification, but I'm going to post this here because it becomes homogeneous, and I know $$v = \frac{y}{x}$$ but I don't see this simplification, I don't understand how it gets from this...

$$\frac{dy}{dx} = \frac{y-x}{y+x}$$

To THIS:

$$= \frac{v-1}{v+1}$$ (I'm just only showing the RHS here)

If someone wouldn't mind explaining it, that would be great because I'm lost, unless this is a special rule.

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cristo
Staff Emeritus
Well, let y=vx, then sub into the RHS and you get $$\frac{vx-x}{vx+x}=\frac{x(v-1)}{x(v+1)}=\frac{v-1}{v+1}$$

oh wow, that's incredible, I get it, thanks cristo

cristo
Staff Emeritus