In This Lesson Definition & Key Transforms Properties Solving DEs with Laplace Step & Impulse Functions 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)
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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))
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Inverse: y(t) = 2e⁻ᵗ − e⁻²ᵗ
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Step & Impulse Functions 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
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.