Partial fractions where the denominator is one function

Question

I need to solve this differential equation for x:

$$ \frac{dv}{dx} = \frac{4000}{v} - 0.9v $$

Rearranging:

$$ \frac{dx}{dv} = \frac{1}{4000v^{-1} - 0.9v} $$

How would I go about splitting this into partial fractions so that I could integrate it?

Answer 1

Indeed, you need to consider the ODE as:

$$\frac{dv}{dx}=\frac{a}{v}-bv=\frac{a-bv^2}{v}$$ and so if $v\neq 0$ then you get: $$\frac{v}{a-bv^2}dv=dx$$