Solve DE for approaching terminal velocity

[magicfountain]

Homework Statement

I'm trying to find the function, that describes the velocity approaching to a terminal velocity.

Homework Equations

$F_{net}=mg-\frac{1}{2}\rho v^2 AC_d$

The Attempt at a Solution

$F=ma$
$a=F/m$
$\dot{v}=F/m=g-\frac{1}{2m}\rho v^2 AC_d$
$\dot{v}=g-kv^2$
$\dot{v}+kv^2=g$

(k and g are constants)
I have very few knowledge of DEs and it seems hard to guess a solution.
Can somebody help me?

 

[Saitama]

Rewrite the equation as
$$\frac{dv}{dt}=g-kv^2$$
$$\frac{dv}{g-kv^2}=dt$$

Now it should be easy to solve.

 

[magicfountain]

thank you!
now it seems obvious. :D

 

[Saitama]

The expression can be rewritten as:
$$\frac{dv}{k((\sqrt{\frac{g}{k}})^2-v^2)}=dt$$

Integrating LHS is same as integrating $\frac{dx}{a^2-x^2}$ where a is some constant. Integrate $\frac{dx}{a^2-x^2}$ using partial fractions.