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.