Where the damping force is given as .
This is a homogeneous linear differential equation so the auxiliary equation is
The solution to this equation is (using the quadratic formula)
So the general solution for the differential equation is
So the nature of the discriminant will determine the nature of the solution.
Your problem specifies m = 1 kg and c = 10 Ns/m. (You didn't specify the units for this, so I'm presuming they are MKS units.)
So your solution comes out to be:
The nature of the solution changes depending on the sign of the discriminant. If the discriminant is negative we have damped harmonic motion. If the discriminant is 0 we have critically damped harmonic motion. If the discriminant if positive we have overdamped harmonic motion.