notes152-ch8

# notes152-ch8 - MATH 152 Spring 2004-05 Applied Linear...

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MATH 152 Spring 2004-05 Applied Linear Algebra & Diﬀerential Equations Lecture Notes Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology April 18, 2005

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Contents Table of Contents iii 1 Introduction 3 1.1 What are Diﬀerential Equations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Classiﬁcation of Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Solutions of Ordinary Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . 5 2 First-Order Differential Equations 11 2.1 First-Order Linear Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Modeling with First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Euler and Runge–Kutta Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 35 3 Second-Order Linear Equations 43 3.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Homogeneous Equations with Real Constant Coeﬃcients . . . . . . . . . . . 46 3.1.2 Method of Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Method of Undetermined Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Method of Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Laplace Transform 81 4.1 Introduction – Examples and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Initial Value Problems and Inverse Laplace Transform . . . . . . . . . . . . . . . . . 86 4.3 Discontinuous Forcing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Impulse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5 Matrix 103 5.1 Introduction and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Addition and Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5 Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.6 Partitioned Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 iii

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CONTENTS 6 Systems of Linear Equations 117 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Solve Systems of Linear Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . 119 6.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 Theory of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3.1 Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3.2 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.3.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.4 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.5 Find Inverse of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7 Euclidean Vector 139 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.1.1 Euclidean Vector and Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 139 7.1.2 Addition and Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 140 7.1.3 Column and Row Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 General Solution in Vector Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.1 Span of One or Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.2 Determine Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.4.1 Independence of One or Two Vectors . . . . . . . . . . . . . . . . . . . . . . . 153 7.4.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4.3 Determine Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.5 Basis of R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8 Eigenvalue and Eigenvector 163 8.1 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.2 Eigenvalue, Eigenvector, Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.3 Find Eigenvalue and Eigenvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3.1 2 × 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3.2 3 × 3 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.3.3 Complex Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 iv
Chapter 8 Eigenvalue and Eigenvector Throughout Chapters 5, 6, 7, we learn basic concepts and techniques in matrix algebra, vectors in Euclidean space, and how to solve systems of linear algebraic equations by use of elementary row operations. In this chapter we shall revisit the world of diﬀerential equations. Again we are interested in solving diﬀerential equations, however, the main diﬀerence from the previous chapters (Chapters 2, 3, 4) is: instead of solving a single diﬀerential equation of one unknown function, we now have a homogeneous system of linear diﬀerential equations in which several unknown functions are mixed up.

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## This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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notes152-ch8 - MATH 152 Spring 2004-05 Applied Linear...

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