The Laplace transform converts time-domain functions to s-domain using an improper integral. The exponential kernel e⁻ˢᵗ ensures convergence for suitable s.
Properties
Linearity: ℒ{af + bg} = aF + bG
Derivative: ℒ{f'(t)} = sF(s) − f(0)
Second derivative: ℒ{f''(t)} = s²F(s) − sf(0) − f'(0)
Shift: ℒ{eᵃᵗf(t)} = F(s − a)
The derivative property is the key insight: differentiation becomes multiplication by s. This turns second-order DEs into algebraic equations in s — much easier to solve!
Solving DEs with Laplace
Example: y'' + 3y' + 2y = 0, y(0) = 1, y'(0) = 0
Transform: s²Y − s − 0 + 3(sY − 1) + 2Y = 0
(s² + 3s + 2)Y = s + 3 → Y = (s + 3)/((s + 1)(s + 2))
The workflow: (1) transform the DE, (2) solve the algebraic equation for Y(s), (3) use partial fractions and the table to invert.
Step & Impulse Functions
Unit step: u(t − a) = 0 for t < a, 1 for t ≥ a
ℒ{u(t − a)·f(t − a)} = e⁻ᵃˢF(s)
Dirac delta: δ(t − a) — impulse at t = a
ℒ{δ(t − a)} = e⁻ᵃˢ
Step functions model sudden switches (turning on a force). The delta function models instantaneous impulses (a hammer strike). These are essential in engineering and signal processing.
The Laplace transform is part of a family of integral transforms. The Fourier transform (using e⁻ⁱωᵗ instead of e⁻ˢᵗ) decomposes signals into frequencies — connecting to trigonometric series. The Z-transform does the same for discrete-time systems.