Solve md²x/dt² + c(dx/dt) - kx = 0 using characteristic equation

[phys2]

Homework Statement

A physical system is designed having the following equation of motion

md²x/dt² + c(dx/dt) - kx = 0.

(a) From the corresponding subsidiary equation, find the solution to this equation of motion. (HINT: use the solution of the damped harmonic oscillator as a guide).
(b) How many distinct types of solution and hence physical behaviour does it exhibit ( does it have solutions that correspond to underdamping, overdamping, critical damping?)

Homework Equations

The Attempt at a Solution

From the hint, I expect the solutions to the system to be similar to the damped solution.

So the damped solution was x = -β +- √ (β² - ω² )

β = c/m and ω² = k/m

So now I am stuck! Anyone care to point me in the right direction?
Thanks!

 

[rock.freak667]

Hint: x=e^rt is a solution of that equation (so substitute it into the DE and see what you get).