In This Lesson Definition & Key Transforms Properties Solving DEs with Laplace Step & Impulse Functions 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 Vgj8E11AiYR10wR4pTivZE7OVqmMwrZeGSpQE9sKZ8TWsRtoSJQUtb1hl3zZEWlcEF9NUwRnCbwrqPuxzEpZCgKrdS4dtNvovUdW2d73h3oyHywDWWzdin+pvRfLitlTb9d4Aogd2siNtEHF3j9tXgis/x+OrVQm5EJJwaY18HS7+KnODLj0vq0jyrKYSFKpt1tI7wdVIDQ/6DPDdqxwdL86Jx/ea4IMB9N2hh22MPm6kzSC3uoO7csuh1cYb0npZkszXp6gEPauLdYlK0dsqbSHhomUuS9pZzYKGs7wjsAlHQScqU+76cFO/rinOaQVV79w70IVKHCdMVeAPyOyktn6srFn28R6c0nJp0aPdSL/I30+ioJ8dg/GtvYSIJrhOUwWdVINoYOi06QtDn/WEe9SEN5ZnHlt8htmQVfYd49SRdOFFf/6j2dU3uYtgj/5XCiPXwekL0tJMXq+pg8ngMa1BaBZG0nEwiYWqZkOc2cP0tk0/mdsIOlL8YtGgow76QYSB9ojS+U9qto0x+G9S805fLFqZUa/YnPaVyKaINotmz9J/xqCEL1JmMURbP6iD3DwzsxdSO1a+LgANgCDeBlA79C3x15qnHqAFA6pRQLxgi0o7Dy3he5xFcAJo4KW+XakYHNPpWMwRDOIrvzACwU1FWxsEr3LCqEJKyZzdHwnXKZQZrcv9gu+IgxU+sQN9i3JeoXpjKWO/8LDuHEXEpVXX4Kmrd0oiliaIqke3qHeZtMND4YFM4bNLBkUZmtVq2r/+BfxY/fFuC4or8djX84jfZ+gBmhyA+JGQej0xgbSrZ9rgru9jNjlPD9o++qqgVs0bLPbh/W+gLkkRY1mNlRrV50tpFYgaiMzJBgrM3tVCNtx/vjpb7Dqa2rFRzxVTrQUYeWoOr7ediHUpu9p18ZdS43f6B2w43w1xg2Yh1AcQzpAhH+z5kW5qs8pd4FiUizNgL4EloV1E9ZyVafJaXt5qNzbwt57Xbd 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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.
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)
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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.