Transform Properties Lab

Differential Equations · Axiom Academy

Four rules turn hard time-domain operations — combining, shifting, scaling, multiplying by t — into simple algebra on F(s) . Watch one, then drive two yourself. The whole toolkit rests on one habit: properties, not re-deriving Every Laplace transform you'll ever need comes from a small set of PROPERTIES applied to a short table of known pairs — you almost never integrate by hand again. Watch the simplest property first: linearity turns "combine the functions" into "combine the transforms." Press play: f(t)=e^ -t and blend into 2f(t)+3g(t) in the time domain, while their transforms and simultaneously blend into 2F(s)+3G(s) in the s-domain — the SAME coefficients, on both sides, at the same time. Linearity: — combine first or transform first, same answer either way. First Shifting: multiplying by e^ at shifts F(s) sideways Drag a . In the time domain, f(t)=t gets multiplied by e^ at , growing or decaying faster. In the s-domain, watch slide horizontally to — multiplying by an exponential in time is just a HORIZONTAL SHIFT of the transform. — confirmed here for f(t)=t : , so . Multiplying by t in time is DIFFERENTIATING F(s) (with a sign flip) Drag the point along the s-axis. The left curve is (the transform of f(t)=e^ -t ); the right curve is . Watch the tangent slope on F(s) — its NEGATIVE is exactly the height plotted on the right. — every extra factor of t costs another derivative (with another sign flip): . Four properties, one algebra toolkit

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