The Transform Method

Differential Equations · Axiom Academy

How the Laplace transform turns derivatives into multiplication by s — collapsing a differential equation into ordinary algebra. Take the Laplace transform of a derivative and it comes back as multiplication by s , minus the starting value . This one substitution is what makes the whole method work: Both formulas come from the same integration-by-parts move — apply it once to get the y' rule, then apply that rule again to y' itself (treating y''=(y')' ) to get the y'' rule: 2. The Complete Solution Process Solving any linear ODE with Laplace transforms is the same four moves every time, regardless of the equation: 3. Working an Initial Value Problem Solve y'' + y = 0 with — a spring or pendulum released from rest at position 1. Watch the differential equation collapse into algebra: Step 1 — Transform both sides: Step 2 — Substitute the derivative formula and the initial conditions : Step 3 — Solve the algebraic equation for Y(s) : Step 4 — Recognize the inverse transform. The transform tables give — here b=1 , so: The transform method turns differentiation into multiplication by s , collapsing any linear ODE into one algebra problem in Y(s) . Scroll up to revisit any step.

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