Help finishing a linear differential equation. Mechanics

[navm1]

Homework Statement

Find the distance which an object moves in time t if it starts from rest and has an acceleration d^2x/dt^2 = ge^-kt.
Show that for small t the result is approx "x=(gt^2)/2" and show that for very large t, the speed is approximately constant. the constant is called the terminal speed.

Homework Equations

The Attempt at a Solution

I ended up with v = -(ge^-kt)/k + v_0 and x = (ge^-kt)/k^2 + v_0t + x_0 however I am not sure what to do next. I have tried to solve for k and also set t as zero to get x = g/k^2 but it doesn't seem to be the answer the book is looking for.

Thanks

 

[RUber]

For small t, look at the Taylor expansion for the exponential.
For large t, take the difference between $\int_0^{T+\Delta} ge^{-kt}\, dt-\int_0^T ge^{-kt}\, dt$ for large T, or instead of $T+\Delta,$ use $\infty$.

 

[vela]

$k$ is a given parameter. You don't want to solve for it.

You might find it helpful to use definite integrals, e.g.,
$$\int_{v_0}^v \,dv = \int_0^t ge^{-kt}\,dt.$$ The problem statement says the object starts from rest, so use that bit of information too.