# Can't find coefficient for my IVP

In my problem I have to set up an IVP and model freefall with air resistance before the bungee starts being pulled on. Beta being my airresistance coefficient. I have: $$mx'' + \beta x' = f(t) = 0$$ $$m = \frac{75}{16} \quad and \quad \beta = 0.5$$ Solving for k in my characteristic equation gives me: $$k^{2} + \frac{8}{75}k = 0$$ $$k_{1}=0 \quad and \quad k_{2}=-\frac{8}{75}$$ Thus my general solution is: $$x(t) = c_{1}+c_{2}e^{-\frac{8}{75}t}$$ $$x'(t) = -\frac{8}{75}c_{2}e^{-\frac{8}{75}t}$$ But trying to find either c with my initial conditions of x'(0) = 0 and x(0) = 0 (it equals zero as I want down to be considered positive in this scenario) I keep getting either both c's are equal to zero or just the second c is, which means the velocity function doesn't work. I'm a bit lost on how to proceed.

If you think about the initial conditions in terms of physics, and connect that to your specific DE, you will see that when the velocity is zero ($$x'(0)=0$$) then your acceleration is zero: $$x''=\frac{\beta}{m}x'.$$