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Laplace Transforms

Convert differential equations into algebra — then transform back.

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Definition & Key Transforms

ℒ{f(t)} = F(s) = ∫₀^∞ e⁻ˢᵗ f(t) dt

ℒ{1} = 1/s  |  ℒ{tⁿ} = n!/sⁿ⁺¹
ℒ{eᵃᵗ} = 1/(s−a)  |  ℒ{sin(bt)} = b/(s²+b²)
ℒ{cos(bt)} = s/(s²+b²)

The Laplace transform converts time-domain functions to s-domain using an improper integral. The exponential kernel e⁻ˢᵗ ensures convergence for suitable s.

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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))

Partial fractions: Y = 2/(s + 1) − 1/(s + 2)

Inverse: y(t) = 2e⁻ᵗ − e⁻²ᵗ

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.

Related Topics

Second-Order DEsLaplace transforms excel at solving 2nd-order IVPs. IntegralsThe Laplace transform is defined as an improper integral. PolynomialsPartial fraction decomposition is key to inverse transforms. Trig ApplicationsFourier analysis decomposes signals into sine/cosine components.