Applying Newton's second law to the force equation, we obtain

$\mathrm{ma}\=-k y - c\frac{\ⅆy}{\ⅆt}\,$

Since we know that acceleration is just the second derivative of position, we can write this as

$m\frac{{\ⅆ}^{2}y}{\ⅆt^2} \=-k \mathrm{y}- c\frac{\ⅆy}{\ⅆt}\,$

or

$\frac{{\ⅆ}^{2}y}{\ⅆt^2} \+2\cdot \mathrm{\γ}\frac{\mathrm{dy}}{\ⅆt}\+{\mathrm{\ω}}^2 y \= 0\,$

with the damping factor $\mathrm{\γ}\=\frac{c}{2 m}\,$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 $\mathrm{\γ}-\mathrm{\ω}$:

When the damping factor matches the natural frequency,  the solution is the sum of decaying exponentials:

$y\left(t\right) \= \left(\mathrm{At}\+ B\right)e^{-\mathrm{\ω} t}\.$

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:

$y\left(t\right) \= {\mathrm{Ae}}^{-\left(\mathrm{\γ}-\sqrt{{\mathrm{\γ}}^2-{\mathrm{\ω}}^2}\right)t}\+ {\mathrm{Be}}^{-\left(\mathrm{\γ}\+\sqrt{{\mathrm{\gamma }}^2-{\mathrm{\omega }}^2}\right)t}\.$

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) \= \(A \sin \left(\(\sqrt{{\mathrm{\ω}}^2-{\mathrm{\γ}}^2} t\right) \+ B \cos \left(\sqrt{{\mathrm{\omega }}^2-{\mathrm{\gamma }}^2} t\right)\) e^{-\mathrm{\γt}}\.$

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.