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Chapter 13 Piecewise Forcing Functions
Real-world systems rarely behave with smooth, unbroken motion. Machines switch on, circuits reset, and forces might act for only a moment before stopping. These situations call for
piecewise functions —functions defined by different rules over different time intervals.
When piecewise functions appear as the “inputs” or forcing terms in a differential equation, the Laplace transform method is still up to the task—but we need one more tool: the
unit step function . This mathematical ON–OFF switch allows us to rewrite piecewise functions into a single, concise expression.
In this chapter, you’ll learn how to express any piecewise function using unit step notation, how to handle different types of switches (turning on, turning off, or staying on for just a window of time), and how to apply special Laplace transform rules for step functions. By the end, you’ll be able to solve differential equations with inputs that start, stop, and change just like the systems they model.
🗝️ Key Takeaways...
Step functions are mathematical ON/OFF switches that let you control when a function is active.
The unit step function \(u(t)\) is given by
\begin{equation*}
u(t) =
\left\{
\begin{array}{ll}
1, \amp t \ge 0 \\
0, \amp t \lt 0
\end{array}
\right.
\end{equation*}
jumps from \(0\) to \(1\) at \(t=0\text{.}\) We say it activates at \(t=0\text{.}\)
To active a function at \(t=0\text{,}\) multiply it by \(u(t)\text{.}\)
You can change the activation point by shifting \(u(t)\) horizontally:
\begin{equation*}
u_c(t) = u(t-c) =
\left\{
\begin{array}{ll}
0, \amp t \lt c \\
1, \amp t \ge c
\end{array}
\right.
\end{equation*}
Multiplying by \(u_c(t)\) activates a function at \(t = c\text{:}\)
\begin{equation*}
f(t)\,u_c(t) = \begin{cases}0,& t \lt c\\ f(t),& t \ge c\text{.}\end{cases}
\end{equation*}
By default, \(u_c(t)\) is an OFF-ON switch, but you can turn it into an ON-OFF switch with:
\begin{equation*}
1-u_c(t) =
\left\{
\begin{array}{ll}
1, \amp t \lt c \\
0, \amp t \ge c
\end{array}
\right.
\end{equation*}
So multiplying by \(1-u_c(t)\) deactivates an active function at \(t = c\text{:}\)
\begin{equation*}
f(t)\,(1 - u_c(t)) = \begin{cases}f(t),& t \lt c\\ 0,& t \ge c\text{.}\end{cases}
\end{equation*}
To activate a function only on a window, multiply:
\begin{equation*}
f(t)\,(u_c(t) - u_d(t)) = \begin{cases}f(t),& c \le t \lt d\\ 0,& \text{otherwise}\text{.}\end{cases}
\end{equation*}
When a differential equation has a piecewise forcing term, start the Laplace method process by rewriting it using step functions.
Combine like step terms to minimize how many Laplace transforms you have to compute.
After rewriting the forcing function, apply the same three Laplace steps you already know: forward transform, solve for \(Y(s)\text{,}\) and invert.
