This is question 1.4 of Chapter 8 of Mary L. Boas's Mathematical Methods in the Physical Sciences, Edition 2. I'm using it as a substitute for my ordinary differential equations class since my textbook has apparently been lost somewhere in the mail. 1. The problem statement, all variables and given/known data Find the distance which an object moves in time t if it starts from rest and has an acceleration d^2/dt^2 = ge^(-kt). Show that for a small t the result is approximately (x = 1/2(gt^2)) and for very large t, the velocity dx/dt is approximately constant. (This problem corresponds roughly to the motion of a parachutist.) 2. Relevant equations The solution manual gives the solution as: x=k^(-1)gt + k^(-2)g(e^(-kt)-1) 3. The attempt at a solution I integrated both sides of d^2/dt^2 = ge^(-kt) twice with respect to t, ending up with x = k^(-2)ge^(-kt) + v_0(t) + x_0. However substituting t=0 into the equation gives x = g/(k^2) + x_0, not (1/2)gt^2, so, obviously on the wrong track. I put the process in an attached image below since it might be easier to read that than my terrible equation formatting. Thanks!