In This Lesson Definition & Key Transforms 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 Properties 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 Solving DEs with Laplace Step & Impulse Functions 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 Definition & Key Transforms 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
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.
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.