Differential equation/rational function

[nos]

Hi all,

Three forces are acting on an object.
Gravity, mg
Lorentz force, av(a is a constant)
Air resistance, kv^2( k is a constant)
v is the velocity of the object

I figured that this differential equation must be the right one:

M(total)\frac{dv}{dt}= mg-av-kv^2

Where m and M(total) are different masses

\frac{dv}{kv^2+av-mg} = -\frac{dt}{M(total)}

Integrate both sides:

-\frac{2}{\sqrt{a^2+4kmg}} artanh(\frac{2kv+a}{\sqrt{a^2+4kmg}}) = -\frac{T}{M(total)} + C

So I want to find the expression for velocity v.
How do I begin? And should the integration constant even be there? When I put limits on the integral(v goes from 0 to v and t goes from 0 to t) then there is no constant of integration.
Thanks so much.

 

[nos]

I'm actually starting to think that maybe the whole anti-derivative is wrong.

 

[nos]

So I figured using the limits on the integral gives me:

arctanh(\frac{2kv+a}{\sqrt{a^2+4kmg}})-arctanh(\frac{0+a}{\sqrt{a^2+4kmg}})=\frac{\sqrt{a^2+4kmg}}{2M(total)}T

Taking the second term arctanh tot the other side and taking the tanh of both sides:

2kv+a = \sqrt{a^ 2+4kmg} tanh(\frac{\sqrt{a^2+4kmg}}{2M(total)}T
+arctanh(\frac{a}{\sqrt{a^2+4kmg}}))

which will finally give v in terms of t:

v(t) = \frac{-a}{2k}+\sqrt{a^ 2+4kmg} tanh(\frac{\sqrt{a^2+4kmg}}{2M(total)}T
+arctanh(\frac{a}{\sqrt{a^2+4kmg}}))/2k