The Laplace Transform is a powerful mathematical tool used to convert a time-domain function into a frequency-domain function. It makes solving differential equations easier by converting them into algebraic equations.
Definition of Laplace Transform
If f(t) is a function defined for t ≥ 0, then the Laplace Transform is:
L{f(t)} = F(s) = ∫₀^∞ e⁻ˢᵗ f(t) dt
Conditions for Existence
- Function must be of exponential order.
- Function must be piecewise continuous.
Standard Laplace Transforms
- L{1} = 1/s
- L{tⁿ} = n!/sⁿ⁺¹
- L{eᵃᵗ} = 1/(s − a)
- L{sin at} = a/(s² + a²)
- L{cos at} = s/(s² + a²)
- L{sinh at} = a/(s² − a²)
- L{cosh at} = s/(s² − a²)
- L{eᵃᵗ sin bt} = b / [(s − a)² + b²]
- L{eᵃᵗ cos bt} = (s − a) / [(s − a)² + b²]
Linearity Property
L{af(t) + bg(t)} = aF(s) + bG(s)
First Shifting Theorem
If L{f(t)} = F(s), then:
L{eᵃᵗ f(t)} = F(s − a)
Second Shifting Theorem
If f(t) is multiplied by a step function u(t − a):
L{f(t − a) u(t − a)} = e⁻ᵃˢ F(s)
Laplace Transform of Derivatives
L{f′(t)} = sF(s) − f(0)
L{f″(t)} = s²F(s) − s f(0) − f′(0)
Laplace Transform of Integrals
L{∫₀ᵗ f(τ) dτ} = F(s)/s
Inverse Laplace Transform
The inverse Laplace Transform is used to get back f(t) from its transform F(s).
L⁻¹{F(s)} = f(t)
Standard Inverse Laplace Transforms
- L⁻¹{1/s} = 1
- L⁻¹{1/s²} = t
- L⁻¹{n!/sⁿ⁺¹} = tⁿ
- L⁻¹{1/(s − a)} = eᵃᵗ
- L⁻¹{s/(s² + a²)} = cos at
- L⁻¹{a/(s² + a²)} = sin at
Unit Step Function (Heaviside Function)
u(t − a) = 0 for t < a and 1 for t ≥ a.
Transform
L{u(t − a)} = e⁻ᵃˢ/s
Convolution Theorem
(f * g)(t) = ∫₀ᵗ f(τ) g(t − τ) dτ
L{f * g} = F(s) G(s)
Partial Fractions Method (Important for Inverse Laplace)
Used when F(s) is a rational function. Split it into simpler fractions and apply known inverse transforms.
Example Type
F(s) = 5/(s² + 4s + 5)
Convert denominator to completed square and apply inverse Laplace rules.
Solving Differential Equations Using Laplace Transform
- Take Laplace transform of both sides.
- Use derivative property.
- Solve algebraic equation in F(s).
- Apply inverse Laplace to get final solution.
Applications
- Electrical circuits (RLC circuits)
- Mechanical vibrations
- Chemical kinetics
- Pharmacokinetic models
- Control systems
Examples (Concept Overview)
- Laplace transform of sin 3t, cos 5t, e⁴ᵗ
- Inverse Laplace using partial fractions
- Using first shifting theorem
- Solving DE like y″ + 3y′ + 2y = 5 using Laplace
Practice Questions
- Find L{t³}
- Find L⁻¹{1/(s² + 9)}
- Find L{eᵗ sin 2t}
- Use partial fractions to find inverse Laplace of 6/(s² + s)
- Solve y′ + 4y = e⁻ᵗ using Laplace transform
Easy-to-understand notes on Laplace Transform for exams and revision.
Detailed Notes:
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