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Chapter 9 Laplace Transform §9.1 Definition of Laplace Transform §9.2 Properties of Laplace Transform §9.3 Convolution §9.4 Inverse Laplace Transform §9.5 Application of Laplace Transform

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§9.1 Definition of Laplace Transform Definition

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Unit Step: Unit Impulse

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Ex:

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TH.9.1.1 Existence Theorem

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Note: The conditions in the theorem are sufficient, not necessary.

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Ex:9.1.5 Ex:9.1.6

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§9.2 Properties of Laplace Transform 1.Linearity Ex.9.2.1

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2.Derivation

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Ex.9.2.2

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Ex.

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3.Integration

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Ex.9.2.4 Ex.

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Homework P217:2.(1)(3)(5) 3 4 5(1)(2)(3)(4)

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O t f(t)f(t) f(t ) 4.Delay Ex: 1 u(t)u(t) tO

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Ex:9.2.8

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Ex:9.2.9

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5.Displacement Ex:

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6.Initial ＆ Terminal Value Theorems (1).Initial Value Theorem

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(2).Terminal Value Theorem

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Ex:9.2.11 Satisfying the conditions of the theorem, then you can use the theorem.

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Ex:9.2.12 Ex:9.2.14 Table for properties on P201

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§9.3 Convolution 1.Definition is called the convolution of,denoted as, i.e.. Note: Convolution in Fourier transform is same to that in Laplace transform.

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Properties: 1.Commutative Law 2.Associative Law 3.Distributive Law 4.

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Ex:

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2.Convolution Theorem TH.9.3.1

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Ex.9.3.2

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Homework: P217:5.(5)-(13) 7.(1)(3)(5) 8

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§9.4 Inverse Laplace Transform 1.Inverse Integral Formula From the inverse Fourier transform, we have the inverse Laplace transform formula.

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R O Real axis Imaginary axis L CRCR +jR jRjR singularities analy TH.9.4.1

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2.Evaluation (1).Using integral formula Ex:

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(2).Using convolution theorem Ex:

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(3).Using partial fraction Ex:

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(4).Using properties Ex: (5).Using L-transform table

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§9.5 Application of Laplace Transform 1.Evaluating the improper integral

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Using Laplace transform solves the differential equation: The block diagram shows the details. Solution of Differential equation Algebra equation of 2.Solving Differential Equation

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Ex:

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Homework: P218: 9.(1)(3)(5) 10.(1)(3) 11.(1)(3)

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1. The properties of Laplace Transform. 2. Application in solving differential equations. The key points and difficulties of the chapter.

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