Real-world systems rarely behave in one smooth, unbroken motion. Machines switch on, circuits reset, and forces might act for only a moment before stopping. These situations call for piecewise functionsโfunctions defined by different rules over different time intervals.
When piecewise functions appear as the โinputsโ or forcing terms in a differential equation, the Laplace transform method is still up to the taskโbut we need one more tool: the unit step function. This mathematical ONโOFF switch lets us rewrite piecewise functions into a single, clean expression.
In this chapter, youโll 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, youโll be able to solve differential equations with inputs that start, stop, and change just like the systems they model.
