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m(d^2x/dt)=-gamma-c(dx/dt) but I can't seem to come up with a method that makes sense. Note: Gamma is just some constant. I tried to integrate both sides wrt t but then I end up with both a velocity function and x(t) in the same equation and that physically doesn't make sense.

I've also tried to turn this into a 2nd order differential equation by saying:

m(d^2x/dt)+c(dx/dt)+gamma=0 . However, now, the issue is that I can't turn this into the characteristic polynomial because gamma does not have an x.

The gist of the problem is that For what Vo must a cup experience in order for it to fall off of a table of length H if the only force acting on the cup is coulumb friction, -gamma. I found that Vo>sqrt(2Hgamma/M). Now I need to solve this same problem when coulomb friction is now -gamma-c(dx/dt).

Any help solving m(d^2x/dt)+c(dx/dt)+gamma=0 or an idea of a different method would be great. thanks!