# Math Help - Separation of Variables question!

1. ## Separation of Variables question!

v (dv/dx) = - (1/250) v² - g

solve the equation for v as a function of x

this is for upward motion using quadratic model of air resistance
thanks

2. Originally Posted by acu04385

v (dv/dx) = - (1/250) v² - g

solve the equation for v as a function of x

this is for upward motion using quadratic model of air resistance
thanks
Is $g$ a constant?

If so...

$v\,\frac{dv}{dx} = -\frac{1}{250}v^2 - g$

$\frac{dv}{dx} = -\frac{1}{250}v - gv^{-1}$

$\frac{dv}{dx} = -\frac{v}{250} - \frac{g}{v}$

$\frac{dv}{dx} = \frac{-v^2 - 250g}{250v}$

$\frac{dv}{dx} = -\frac{v^2 + 250g}{250v}$

$\frac{dx}{dv} = -\frac{250v}{v^2 + 250g}$

$\frac{dx}{dv} = -250v(v^2 + 250g)^{-1}$.

You can now solve this with a $u$ substitution.