Laplace Transforms
Laplace Transforms
Gilles Cazelais May 2006
Contents
1 Problems 1.1 Laplace Transforms . . . . . . 1.2 Inverse Laplace Transforms . 1.3 Initial Value Problems . . . . 1.4 Step Functions and Impulses 1.5 Convolution . . . . . . . . . . 2 Solutions 2.1 Laplace Transforms . . . . . . 2.2 Inverse Laplace Transforms . 2.3 Initial Value Problems . . . . 2.4 Step Functions and Impulses 2.5 Convolution . . . . . . . . . .
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A Formulas and Properties 41 A.1 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . 41 A.2 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . 42 A.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . 43 B Partial Fractions 45 B.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 B.2 Cover-up Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Bibliography 49
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CONTENTS
Chapter
1
(d) f (t) = e−2t (4 cos 5t + 3 sin 5t) (e) f (t) = t3 e2t + 2te−t (f ) f (t) = (1 + e3t )2
Problems
1.1 Laplace Transforms
1. Find the Laplace transform of the following functions. (a) f (t) = 4t2 − 2t + 3 (b) f (t) = 3 sin 5t − 2 cos 3t (c) f (t) = 3e2t + 5e−3t
2. Use an appropriate trigonometric identity to find the following Laplace transforms. See page 43 for a list of trigonometric identities. (a) L cos2 t (b) L {sin 2t cos 2t} (c) L {sin 3t cos 4t} (d) L {sin(ωt + φ)} (e) L {cos(ωt + φ)} (f ) L e−2t
Gilles Cazelais May 2006
Contents
1 Problems 1.1 Laplace Transforms . . . . . . 1.2 Inverse Laplace Transforms . 1.3 Initial Value Problems . . . . 1.4 Step Functions and Impulses 1.5 Convolution . . . . . . . . . . 2 Solutions 2.1 Laplace Transforms . . . . . . 2.2 Inverse Laplace Transforms . 2.3 Initial Value Problems . . . . 2.4 Step Functions and Impulses 2.5 Convolution . . . . . . . . . .
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A Formulas and Properties 41 A.1 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . 41 A.2 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . 42 A.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . 43 B Partial Fractions 45 B.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 B.2 Cover-up Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Bibliography 49
i
ii
CONTENTS
Chapter
1
(d) f (t) = e−2t (4 cos 5t + 3 sin 5t) (e) f (t) = t3 e2t + 2te−t (f ) f (t) = (1 + e3t )2
Problems
1.1 Laplace Transforms
1. Find the Laplace transform of the following functions. (a) f (t) = 4t2 − 2t + 3 (b) f (t) = 3 sin 5t − 2 cos 3t (c) f (t) = 3e2t + 5e−3t
2. Use an appropriate trigonometric identity to find the following Laplace transforms. See page 43 for a list of trigonometric identities. (a) L cos2 t (b) L {sin 2t cos 2t} (c) L {sin 3t cos 4t} (d) L {sin(ωt + φ)} (e) L {cos(ωt + φ)} (f ) L e−2t