pascoetrott on 03:54:43 2005-09-23 asked:
In order to solve the equation of motion of a falling parachute I need to do an
integration that I dont know how to tackle. I would be grateful to any
who knows how this is solved! Here is the equation. d2y/dt2 = g - k *
(dy/dt)^2 where t = time, y = vertical fall, g = gravity, k = constant

You might find it easier to let dy/dt = x
Then, your equation simplifies to:

  dx/dt = g - k * x^2

We can see that:
 d(x^2)/dt = 2 * x * dx/dt
This is a first order ordinary differential equation, with an integrating factor:

  u = e^(integral(

oops sorry accidentally put that last one up too quickly, ignore the
part about the ordinary differential equation
from dx/dt = g - k*x^2 we can see:

 dx/(g-k*x^2) = dt

If you integrate both sides, now, you get ArcTanh(x*sqrt(k/a))/sqrt(k*a) = t
solving in terms of x:

x = sqrt(a/k) * Tanh(t*sqrt(k*a))

which means:

y = integral(sqrt(a/k) * Tanh(t*sqrt(k*a)))
  = 1/k * ln(cosh(x * sqrt(a*k)))


http://integrals.wolfram.com is a handy website for stuff like this
you'll have to check my math, it's hard doing this without a pen and paper

jastsui's answer is correct in principle but there are some typos:

(i) Replace a with g 
(ii) In the final expression, replace x with t

i solved this problem and u can get this from

http://www.geocities.com/satheeshphysics/Solution1.pdf
http://www.geocities.com/satheeshphysics/Solution2.pdf