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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.

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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: $$x''=\frac{\beta}{m}x'.$$

If the acceleration and the velocity are zero, the system won't change position, based on your DE.

You need a new DE, or a new initial velocity.

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