In This Lesson Definition & Key Transforms Properties Solving DEs with Laplace Step & Impulse Functions 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 0BBdMPGABxBrv/Zgf5iRyH5T/pdFelWFN1e2I7pm+dt+3KA1+l4ezdu99iwMhz7ewKw3zbzKrOhle624N0jaP/CgXO8vFnlOK2TKDHJD1Ysr9mR/Ht2gtvET0+LsNMUI78wzAUe4FvMuM9pwJV+czB3fd8cZPcgj0N0TUwJCXYRJBrcK7o2aiBlpiPqTCwKVtJD0Mw5oMkjFoVvLIjTAL/OzuywszTtIqexqb5GwHCI+TH1GPd0t/SNGYh0hDBr7PVr3dFb8EHfeHooZRe12f3mXhKNz6WOfHcm7lIeD6r2TLwGO3+qqog4u2ttcJt9GP6pn71QJ9wXtoMOF/dPJoxHXCGluphhb2oyXFc2htAvdgk6VLl460V7E5ppKdG8ow9lr/tTWKZhhtN0Tb1h2wOKkqhl/BOLU7ZD4MS6/1mMuhS5NriruWzswziECUV+/8YOM/vI7IeuHhwMOn53KqhPDGiO4lcJVJyjNF3qSGxEeR65WpPluYVQQ613Q1ol32snojPq1bwy04tc21rVT2fTgwRq98LFZIOwJzQHQLfDTPLissyBNXyBXaNqnCSTDKrcj6YunvGB57bpVjSrgoGaaImdU0FqC97KofiUC5MMqt7fPcq5PHgRfkn+XBbIk55RP/UlLtBE0DxBc2Akmf3mWhEZae8xWS4OWf5fofxeRthKz1+qqoAyfOCvEAQPBA6pn7JdXpS/U53aaSnWvyZr7ETE74FZTOzLOjg73s1ib+5xD2yuyv8K756KihMhn5WsZHdKgGXgd1EFNvGup+Gcqivp+QSFjhjkyKARG9Yz3LXhKDcNFxa6+Z0TyUivVN2Hl863H+tNwt9jCtNrALBIo//KY8uD5BIudMfNq8tyMPAGsoHMx1qzeewltjLwXtMX0GizjzQjD71hNeCcY9bpxF1ApZVFHk9VWLcHcfHErO/ESqNOt3D+NVDmWJenQVIU2eGEDlyXVvgltR 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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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.
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