Appearance of the Jerk Term in Dynamics of Mass-Spring-Damper System I am coming from the computer science territory and have not a long trace in mechanics. My background in derivation of the system dynamics could be summarized with utilization of the Lagrange Mechanics but here is a vague point, with which I have confronted, recently. How does the existence of the the jerk term in dynamics of a Mass-Spring-Damper could be justified, where the Lagrange Dynamics just computes up to the 2nd derivatives of the generalized parameters?!...
If one considers a dynamic system, which (from left to right) consists of a spring with constant $k_1$, a mass $m$, a damper with constant $c$ and the other spring with constant $k_2$, all connected together, respectively, the derived dynamics of the system would be declared as: (The junction of spring $k_2$ and damper $c$ is massless)
$$
\frac{cm}{k_2} · \frac{d^{3} x}{d t^{3}} +
m  \frac{d^{2} x}{d t^{2}}   +
c \left( 1 +\frac{k_1}{k_2}\right) \frac{dx}{dt}+
k_1 x= 0
$$
Obviously, the Jerk term has been appeared, up there, noticeably. 
Would you please guiding me that how such dynamics could be interpreted by either Lagrange Dynamics or Newtonian Method?!
 A: 
If one considers a dynamic system, which (from left to right) consists
  of a spring with constant k1, a mass m, a damper with constant c and
  the other spring with constant k2, all connected together,
  respectively

If I understand your setup correctly, the damper is connected between the mass and the 2nd spring.
Denote the extension of spring 1 with $x_1$ and the compression of spring 2 with $x_2$ (a positive $x_1$ results in a leftward force and likewise for $x_2$).
Since the connection of the damper and spring 2 is massless, the net force there must vanish:
$$0 = -k_2x_2 - c(\dot x_2 - \dot x_1) \Rightarrow c\dot x_2 + k_2x_2 = c\dot x_1$$
At the junction of the mass and spring 1 we have
$$m\ddot x_1 = -k_1x_1 - c(\dot x_1 - \dot x_2) \Rightarrow m\ddot x_1 + c\dot x_1 + k_1x_1 = c\dot x_2$$
Substitute the 1st equation into the 2nd to find
$$m\ddot x_1 + k_1x_1 = -k_2x_2$$
Thus
$$c\dot x_2 = -\frac{cm}{k_2}\dddot x_1 - c\frac{k_1}{k_2}\dot x_1$$
Finally, substitute this into the 2nd equation to yield a third order equation of motion in $x_1$:
$$\frac{cm}{k_2}\dddot x_1 + m\ddot x_1 +c\left(1 + \frac{k_1}{k_2}\right)\dot x_1 + k_1 x_1 = 0$$

A Lagrangian for the system is
$$L = \frac{1}{2}m\dot x^2_1 - \frac{1}{2}k_1x^2_1 - \frac{1}{2}k_2x^2_2$$
The Lagrange equations (of the first kind) are
$$\frac{d}{dt}\frac{\partial L}{\partial \dot x_i} -  \frac{\partial L}{\partial x_i} + \frac{\partial D}{\partial \dot x_i} = 0$$
where
$$D = \frac{c}{2}(\dot x_1 - \dot x_2)^2$$
yielding two equations of motion
$$m\ddot x_1 + k_1x_1 + c(\dot x_1 - \dot x_2) = 0$$
$$k_2x_2 - c(\dot x_1 - \dot x_2) = 0$$
which match the first and second equations above.
A: I'm not sure where the term with the third derivative came from; taking the equilibrium point of our system to be $x=0$ we find the force due to the leftmost spring is given by:
\begin{equation}
F_{\textrm{spr,1}}\left(x\right)=-k_1x
\end{equation}
the force due to the rightmost spring is given by:
\begin{equation}
F_{\textrm{spr,2}}\left(x\right)=k_2x
\end{equation}
and the force due to the damper is given by:
\begin{equation}
F_{\textrm{damp}}\left(x\right)=-c\frac{\textrm{d}x}{\textrm{d}t}
\end{equation}
where $x$ is the position of the mass. We therefore find from Newton's laws that:
\begin{equation}
\begin{aligned}
\sum_i\textbf{F}_i&=m\frac{\textrm{d}^2x}{\textrm{d}t^2}\\
\implies k_2x-k_1x-c\frac{\textrm{d}x}{\textrm{d}t}&=m\frac{\textrm{d}^2x}{\textrm{d}t^2}\\
\implies m\frac{\textrm{d}^2x}{\textrm{d}t^2}+c\frac{\textrm{d}x}{\textrm{d}t}+\left(k_1-k_2\right)x&=0
\end{aligned}
\end{equation}
which has no terms with a third derivative. In fact, this can be solved exactly using the method described here to obtain that:
\begin{equation}
x\left(t\right)=A\textrm{e}^{-\frac{ct}{2m}}\cos\left(\sqrt{\frac{k}{m}-\frac{c^2}{4m^2}}-\phi\right)
\end{equation}
where $A$ and $\phi$ are constants that depend on the initial conditions of our system.
