How Do You Find x(t) for a Particle Under Force F(t) = mae^(-bt)?

[phys2]

Homework Statement

A particle of mass m is subject to the force; F(t) = mae^(-bt). The initial position and speed are zero. Find x(t)

Homework Equations

F = ma

The Attempt at a Solution

So I used F = m d²x/dt²

I then rearranged the formula and got d²x/dt2 = F/m

Substituting F = mae(-bt) into the above equation, I got d2x / dt2 = ae^(-bt)

Then I integrated once but the thing is since I am integrating ae^(-bt) with respect to time, would my limits of integration be t and 0? I assumed it would and got dx/dt = -(a/b)e^-bt + v

Then I integrated again and got x(t) = (a/b^2)e^(-bt) + vt + x

But since the question said v = x = 0 (my initial position and speed), so my final answer should be x(t) = (a/b^2)e^(-bt). Is that right? Thanks!

 

[ehild]

Then I integrated again and got x(t) = (a/b^2)e^(-bt) + vt + x

But since the question said v = x = 0 (my initial position and speed), so my final answer should be x(t) = (a/b^2)e^(-bt). Is that right?

It is right.

ehild