Applying Newton's second law to the force equation, we obtain
$\mathrm{ma}\=kyc\frac{DifferentialD;y}{DifferentialD;t}comma;$
Since we know that acceleration is just the second derivative of position, we can write this as
$m\frac{{\ⅆ}^{2}y}{\ⅆ{t}^{2}}equals;k\mathrm{y}c\frac{DifferentialD;y}{DifferentialD;t}comma;$
or
$\frac{{\ⅆ}^{2}y}{\ⅆ{t}^{2}}plus;2\cdot \mathrm{gamma;}\frac{\mathrm{dy}}{DifferentialD;t}plus;{\mathrm{omega;}}^{2}yequals;0comma;$
with the damping factor $\mathrm{\γ}\=\frac{c}{2m}comma;$and the natural frequency $\mathrm{\ω}\=\sqrt{\frac{k}{m}}\.$The solution to this differential equation can be expressed in one of three ways, depending on the sign of $\mathrm{\γ}\mathrm{\ω}$:
1.

Critically damped, γ = ω.

When the damping factor matches the natural frequency, the solution is the sum of decaying exponentials:
$y\left(t\right)equals;\left(\mathrm{At}plus;B\right){e}^{\mathrm{omega;}t}period;$
Critical damping is desirable for virtually all applications of oscillatory motion as the solution decays the quickest.
The solution can be expressed as a sum of decaying exponential functions:
$y\left(t\right)equals;{\mathrm{Ae}}^{\left(\mathrm{gamma;}\sqrt{{\mathrm{gamma;}}^{2}{\mathrm{omega;}}^{2}}\right)t}plus;{\mathrm{Be}}^{\left(\mathrm{gamma;}plus;\sqrt{{\mathrm{\gamma}}^{2}{\mathrm{\omega}}^{2}}\right)t}period;$
The larger the value of $\mathrm{\γ}$, the slower this solution will decay, due to the dominating exponential term ${\mathrm{Ae}}^{\left(\mathrm{\gamma}\sqrt{{\mathrm{\gamma}}^{2}{\mathrm{\omega}}^{2}}\right)t}$.
In this case the motion is still oscillatory with a decaying amplitude. This is usually the least desirable solution for mechanical systems such as car suspension. The formal solution is
$y\left(t\right)equals;lpar;A\mathrm{sin}\left(lpar;\sqrt{{\mathrm{omega;}}^{2}{\mathrm{gamma;}}^{2}}t\right)plus;B\mathrm{cos}\left(\sqrt{{\mathrm{\omega}}^{2}{\mathrm{\gamma}}^{2}}t\right)rpar;{e}^{\mathrm{gamma;t}}period;$
In this case, the smaller the value of $\mathrm{\γ}$, the slower this solution will decay.
In all cases, the constants A and B are determined from the initial conditions of the problem.