# Using the inverse hyperbolic tangent function to solve ODE

1. Nov 16, 2009

### bitrex

1. The problem statement, all variables and given/known data
Hi all. I have to solve the differential equation $$\frac{dv}{dt} = g(1 - \frac{\rho}{g}v^2)$$.

3. The attempt at a solution

Apparently the solution should involve the inverse hyperbolic tangent function - with the equation in this form it should just be separable, correct? However, when separating variables I have to integrate the function $$\frac{dv}{1-\frac{\rho}{g}v^2}$$ which I am not sure how to go about. I think a substitution of some kind? Any tips would be appreciated.

2. Nov 16, 2009

### HallsofIvy

Staff Emeritus
Partial fractions: $1-\frac{\rho}{g}v^2= (1- \sqrt{\frac{\rho}{g}}v)(1+ \sqrt{\frac{\rho}{g}}v)$.

3. Nov 16, 2009

### Dick

Or v=sqrt(g/rho)*tanh(u), if you want to stick with the hyperbolic function approach. You'll get the same answer, though it will look different.

4. Nov 16, 2009

### bitrex

I see it now. Thanks guys!