Transform Visualization

Differential Equations · Axiom Academy

Watch the Laplace transform actually happen: an exponential kernel weighs down a time-domain function, and the area it leaves behind becomes a single number, F(s) . Differential equations describe how things change in time — but calculus with derivatives and integrals can get messy fast. The Laplace transform offers an escape hatch: multiply your function by a decaying exponential e^ -st , add up the result from t=0 to , and the whole function collapses into one number F(s) . Watch that collapse happen. Below, f(t)=e^ -t is being weighed down by the kernel e^ -st at s=1 . As the sweep moves left to right, the shrinking gold curve is the product , and the teal fill is the area accumulating beneath it — that running total is F(s) . At s=1 , the running total settles at — exactly the closed-form transform of e^ -t . Slide s : watch the weight — and the transform — change The kernel e^ -st is a dial. Turn s up and it clamps down hard on f(t) early, so almost no area survives — F(s) shrinks. Turn s down and more of the function's tail counts, so F(s) grows. Drag the slider and watch the weighted curve and the running total move together. for every s you try here — check a few values against the readout. Every function gets its own transform Pick a time-domain function on the left. Its Laplace transform F(s) appears on the right — a completely different-looking curve, but one that carries all the same information.

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