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# notes152-ch5 - 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 March 14, 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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Chapter 5 Matrix 5.1 Introduction and Examples The modern theory of matrices was discovered and developed in the eighteenth and nineteenth centuries. Initially, their development dealt with transformation of geometric objects and solution of systems of lin- ear equations. Matrices provide a theoretically and practically useful way of approaching many types of problems including: Solution of Systems of Linear Equations, Equilibrium of Rigid Bodies (in physics), Graph Theory, Theory of Games, Economics Model, Computer Graphics, Genetics, Cryptography, Electric Networks, Fractals, etc. Matrices are a very important tool in expressing and discussing problems which arise from real life cases. Let us first consider some simple examples. ¥ Example 5.1.1 (Electric network) In the following electric network, - 20V + - 11V + i 1 i 2 i 3 P Q Figure 5.1: An electric network the currents i 1 , i 2 , i 3 satisfy the equations Node P : - i 1 + i 2 - i 3 = 0 , Node Q : i 1 - i 2 + i 3 = 0 , Left loop : 5 i 1 +2 i 2 = 20 , Right loop : 2 i 2 +3 i 3 = 11 . The above system of equations is characterized by the following rectangular array of numbers £ A b / = - 1 1 - 1 0 1 - 1 1 0 5 2 0 20 0 2 3 11 , called the augmented matrix . 2 103

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5. Matrix ¥ Example 5.1.2 (Rotation in plane) Consider the rotation by an angle θ in the plane with respect to the origin. The rotation maps a point v = ( x, y ) to another point ¯ v = (¯ x, ¯ y ). θ v = ( x, y ) ¯ v = (¯ x, ¯ y ) Figure 5.2: Rotation by angle θ ¯ v may be expressed in terms of v by the formula ( ¯ x = x cos θ - y sin θ, ¯ y = x sin θ + y cos θ.
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notes152-ch5 - MATH 152 Spring 2004-05 Applied Linear...

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