Chapter 4.P3, Problem 4P

In the case that the damping constant $\gamma >0$, find the critical points of the differential equation in Problem $2$ and discuss their stability properties.

The Heaviside, or unit step function, is defined by

$u\left(x-c\right)=\{\begin{array}{c}1,x\ge c\\ 0,x<c.\end{array}$

Use the unit step function to express the differential equation in Problem $1$ in a single line.

Assuming that both springs have spring constant $k$ and that there is a damping force proportional to velocity $x^{\prime }$ with damping constant $\gamma$, write down the differential equation that describes the motion of the mass. Note that the expression for the force function depends on which of the three different parts of its domain $x$ lies within.