Operational Properties

Differential Equations · Axiom Academy

Six rules that turn calculus on f(t) into algebra on F(s) — the toolkit that makes the Laplace method work. The transform is a linear operator : it distributes over sums and pulls constants straight through. This is what lets us break any combination of table-functions into pieces we already know. 2. First Shifting Theorem (s-shift) Multiplying f(t) by e^ at in the time domain slides the whole transform sideways in s — replace every s in F(s) with (s-a) . Find . Start from , then shift : 3. Second Shifting Theorem (t-shift) Delaying f(t) by a seconds — and switching it OFF before then with the unit step u(t-a) — multiplies the transform by e^ -as . This is the rule behind switches, delays, and piecewise-defined forcing. Find — the unit step turned on at t=3 . Here f(t)=1 , so , and a=3 : 4. Derivative of the Transform (multiplying by t^n ) Multiplying f(t) by t^n in the time domain corresponds to differentiating F(s) n times with respect to s — with an alternating sign out front. Running f through a "total so far" integral from 0 to t corresponds, on the s -side, to simply dividing by s . Integration in time is division in the transform domain — the mirror image of the derivative rule. 6. The Derivative Property — Why This Solves DEs

This is the written version of the interactive lesson above. See the full Differential Equations course.