Chapter 4.P3, Problem 3P

Is the differential equation derived in Problems $1$ and $2$ linear or nonlinear ? Explain why.

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.

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.