Chapter 3: Laplace Transforms

Every input in the last chapter was a smooth, everlasting sinusoid. Real machinery is switched: a relay closes, a thermostat clicks, a load lands on a beam at one instant and leaves at another. At the moment of switching the forcing jumps, and the methods built so far turn awkward — undetermined coefficients wants a single formula valid for all time, and a signal that is "nothing, then suddenly something" refuses to be one formula. This chapter's instrument, the Laplace transform $F(s) = \int_0^\infty e^{-st} f(t)\, dt$, is built for exactly this world: it integrates the entire history of a signal, jumps and all, into one well-behaved function of a new variable $s$.

The transform's bargain has three clauses, and together they turn switching into algebra. Differentiation becomes multiplication by $s$, so a differential equation in $t$ becomes a polynomial equation in $s$ — calculus traded for arithmetic. A delay of $a$ becomes the factor $e^{-as}$, so "the same response, started later" costs one exponential tag. And periodic switching — on, off, on, off, forever — becomes a geometric series that sums in closed form. One solves the algebra in $s$, then returns to time by partial fractions; the jumps that broke the old methods ride through as harmless factors.

Two sections carry the chapter. The first throws a single switch: the Heaviside step function, the rest-to-rest response of a damped spring to a force that turns on at $t = a$, and the delay rule that moves the whole response — overshoot, ringing, and all — to any starting time. The second switches forever: the square wave, taken apart two ways, as a train of delayed steps and as a ladder of odd harmonics, with a treacherous payoff — a switched forcing can resonate a system whose frequency it does not appear to contain. The figures are interactive as always: one slider that carries an entire ringing response rigidly through time, and a set of visibility toggles that assemble a square wave out of sinusoids before your eyes. Predict before you drag.