Chapter 1- 6 Orange Book of Lecture notes

# Chapter 1- 6 Orange Book of Lecture notes - ORDINARY...

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Unformatted text preview: ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS — THE LECTURE NOTES FOR MATH- 263 (2010-Fall) ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS JIAN-JUN XU Department of Mathematics and Statistics, McGill University Kluwer Academic Publishers Boston/Dordrecht/London Contents 1. INTRODUCTION 1 1 Definitions and Basic Concepts 1 1.1 Ordinary Differential Equation (ODE) 1 1.2 Solution 1 1.3 Order n of the DE 1 1.4 Initial Conditions 2 1.5 First Order Equation and Direction Field 2 1.6 Linear Equation: 2 1.7 Homogeneous Linear Equation: 3 1.8 Partial Differential Equation (PDE) 3 1.9 General Solution of a Linear Differential Equation 3 1.10 A System of ODE’s 4 2 The Approaches of Finding Solutions of ODE 5 2.1 Analytical Approaches 5 2.2 Numerical Approaches 5 2. FIRST ORDER DIFFERENTIAL EQUATIONS 7 1 Linear Equation 7 1.1 Linear homogeneous equation 8 1.2 Linear inhomogeneous equation 8 2 Nonlinear Equations (I) 10 2.1 Separable Equations. 10 2.2 Logistic Equation 12 2.3 Fundamental Existence and Uniqueness Theorem 14 2.4 Bernoulli Equation: 16 2.5 Homogeneous Equation: 16 v vi ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS 3 Nonlinear Equations (II)— Exact Equation and Integrating Factor 18 3.1 Exact Equations. 18 4 Integrating Factors. 20 3. N-TH ORDER DIFFERENTIAL EQUATIONS 23 1 Introduction 23 2 (*)Fundamental Theorem of Existence and Uniqueness 24 2.1 Theorem of Existence and Uniqueness (I) 24 2.2 Theorem of Existence and Uniqueness (II) 25 2.3 Theorem of Existence and Uniqueness (III) 25 3 Linear Equations 25 3.1 Basic Concepts and General Properties 25 3.1.1 Linearity 26 3.1.2 Superposition of Solutions 26 3.1.3 ( ∗ ) Kernel of Linear operator L ( y ) 27 3.2 New Notations 27 4 Basic Theory of Linear Differential Equations 27 4.1 Basics of Linear Vector Space 28 4.1.1 Dimension and Basis of Vector Space 28 4.1.2 Linear Independency 29 4.2 Wronskian of n-functions 30 4.2.1 Definition 30 4.2.2 Theorem 1 31 4.2.3 Theorem 2 32 4.2.4 The Solutions of L [ y ] = 0 as a Linear Vector Space 33 5 Solutions for Equations with Constants Coeﬃcients 34 5.1 The Method with Undetermined Parameters 34 5.1.1 Basic Equalities (I) 34 5.1.2 Cases (I) ( r 1 > r 2 ) 35 5.1.3 Cases (II) ( r 1 = r 2 ) 36 5.1.4 Cases (III) ( r 1 , 2 = λ ± i µ ) 38 5.2 The Method with Differential Operator 39 5.2.1 Basic Equalities (II). 39 5.2.2 Cases (I) 40 5.2.3 Cases (II) 40 5.2.4 Cases (III) 41 5.2.5 Summary 41 5.2.6 Theorem 1 42 Contents vii 6 Finding a Particular Solution for Inhomogeneous Equation 46 6.1 The Annihilator and the Method of Undetermined Constants 46 6.1.1 The Annihilators for Some Types of Functions 47 6.2 The Method of Variation of Parameters 52 6.3 Reduction of Order 56 7 Solutions for Equations with Variable Coeﬃcients 59 7.1 Euler Equations 59 7.2 The method with differential operators 59 7.3 The method with undetermined parameters 61 7.4 (*) Exact Equations 64 4. LAPLACE TRANSFORMS 67 1 Introduction 67 2 Laplace Transform 69 2.1 Definition 69 2.1.1 Piecewise Continuous Function...
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Chapter 1- 6 Orange Book of Lecture notes - ORDINARY...

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