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Laplace Transform
Course Material
Gyeongsang National University
Dept. of Information & Communication Engineering
Laplace Transform
• Linear differential equation (initial value problem) with constant coefficients can be solved by Laplace transform
Differential Equation Algebraic Equation Differential Equation Solution Algebraic Equation Solution Laplace transform Inverse Laplace transform
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6.1 Laplace transform. Linearity.
s-Shifting
Definition of Laplace Transform
• 𝑓(𝑡) is a function of time
• For 𝑡 ≥ 0, the Laplace transform of 𝑓(𝑡) is defined by following equation:
• 𝐹 𝑠 is a function of 𝑠
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Examples of Laplace Transform
1) 𝑓 𝑡 = 1, 𝑡 ≥ 0
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Example
1) 𝑡
3
2) sin 4𝑡
3) 𝑒
−2𝑡
Linearity
• Linearity
• Example) 1) 3 + 2𝑡
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Shifting Theorem
• s-shifting theorem
• Example) When the Laplace transform of 𝑓(𝑡) is 𝐹 𝑠 = 2𝑠+1
𝑠(𝑠+1), find
the Laplace transform of the following functions. 1) 𝑒−2𝑡𝑓(𝑡)
6.2 Transforms of Derivatives and
Integrals. ODE
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Laplace Transform of Derivatives
• Laplace transform of derivatives
• Example) When the Laplace transform of y(𝑡) is Y(𝑠) and, y 0 = 3, y′ 0 = 1, find the Laplace transform of followings.
1) y′ 2) y′′
Inverse Laplace Transform
• When 𝐹(𝑠) is the Laplace transform of 𝑓(𝑡), 𝑓(𝑡) is an inverse Laplace transform of 𝐹(𝑠), and expressed as follows:
• Example) 1) 2
𝑠3 𝑠
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Inverse Laplace Transform
• Finding an inverse Laplace transform using a partial fraction
– Expressed in partial fraction and then find an inverse Laplace transform of the partial fraction
• Example) 1) 4𝑠−1
𝑠2−𝑠
2) 6𝑠+8
Solving Initial Value Problem
• Solve following initial value problem by using Laplace transform.
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Solving Initial Value Problem
• Solve following initial value problem by using Laplace transform.
Solving Initial Value Problem
• Solve following initial value problem by using Laplace transform.
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Example
• Solve following initial value problem by using Laplace transform.