Differential Equations and Linear Algebra, 4th Edition

10.5 Periodic and Piecewise Continuous Input Functions

Mathematical models of mechanical or electrical systems often involve functions with discontinuities corresponding to external forces that are turned abruptly on or off. One such simple on–off function is the unit step function that we introduced in Section 10.1. Recall that the unit step function at $t=a$ is defined by

$$u_a(t)=u(t-a)=\{\begin{array}{ll}0& \text{if }t<a,\\ 1& \text{if }t\geqq a.\end{array}$$ (1)

The notation $u_a(t)$ indicates succinctly where the unit upward step in value takes place (Fig. 10.5.1), whereas $u(t-a)$ connotes the sometimes useful idea of a “time delay” a before the step is made.

In Example 8 of Section 10.1 we saw that if $a\geqq 0\text{,}$ then

FIGURE 10.5.1.

The graph of the unit step function at $t=a.$

$$\mathcal{L}\{u(t-a)\}={\displaystyle \frac{e^{-as}}{s}}.$$ (2)

Because ...