5.3

Math 3331 Differential Equations5.3 The Inverse Laplace Transform

Jiwen He

Department of Mathematics, University of Houston

jiwenhe@math.uh.edumath.uh.edu/∼jiwenhe/math3331

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

5.3 The Inverse Laplace Transform

Basic Definition

Uniqueness TheoremL-Transform PairsDefinition of the Inverse Laplace TransformTable of Inverse L-Transform

Worked out Examples from Exercises:

2, 4, 6, 7, 9, 11, 14, 15, 17

Partial Fractions

Inverse L-Transform of Rational FunctionsSimple Root: (m = 1)Multiple Root: (m > 1)Examples

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Uniqueness Theorem

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

L-Transform Pairs

F (s) = L(f )(s) = L{f (t)}(s), s > a

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Definition of the Inverse Laplace Transform

F = L(f ) ⇔ f = L−1(F )

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Table of Inverse L-Transform

F = L(f ) ⇔ f = L−1(F )

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.2

2. Compute the inverse Laplace transform of Y (s) = 13−5s .

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.4

4. Compute the inverse Laplace transform of Y (s) = 5ss2+9

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.6

6. Compute the inverse Laplace transform of Y (s) = 23s4

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.7

7. Compute the inverse Laplace transform of Y (s) = 3s+2s2+25

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.9

9. Compute the inverse Laplace transform of Y (s) = 13−4s + 3−2s

s2+49

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.11

11. Compute the inverse Laplace transform of Y (s) = 5(s+2)3

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.14

14. Compute the inverse Laplace transform of Y (s) = 4(s−1)(s−1)2+4

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.15

15. Compute the inverse Laplace transform of Y (s) = 2s−3(s−1)2+5

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Exercise 5.3.17

17. Compute the inverse Laplace transform of Y (s) = 3s+2s2+4s+29

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Inverse L-Transform of Rational Functions

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Simple Root: (m = 1)

Simple Root: (m = 1)

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Multiple Root: (m > 1)

Multiple Root: (m > 1)

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 1

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 1(cont.)

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 2

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 3

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 4

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 5

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 6

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5.3 Basic Definition In-Class Exercises Partial Fractions Examples Examples

Example 7

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