Help rearranging an equation

Question

The original equation is $V + X(dv/dx) = (v^2-4)/(1-v)$

I need to be able separate this so I can solve it (as an ODE)

The intermediate step is listed as $x^{-1} - {(1-v)/(v^2-4)} dv/dx = 0$

I'm not seeing the algebra here.

Answer 1

If the capital $V,X$ are the same as small $v,x$, then simply take $v$ from LHS to RHS to obtain:

$x\frac{dv}{dx}=\frac{2v^2-v-4}{1-v}$, i.e. $\frac{(1-v)dv}{2v^2-v-4}=\frac{dx}{x}$

Now it is separated.