MATH2010 Analysis of Ordinary Differential Equations WORKBOOK Semester 1, 2019 These lecture notes belong to: I can be contacted via: If you find them, please return them to me! School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Australia.
CONTENTS 2 Contents 1 Differential Equations 5 1.1 Introduction 5 1.1.1 Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Systems of ODE's 1.1.4 Texts . 6 7 1.1.3 Laplace Transforms 9 9 1.2 Introduction to Systems of ODE's and Classification of ODE's. 10 1.2.1 Introduction to Systems of ODE's 1.2.2 Classifying ODE's: Linear, Order, Homogeneous. . . . . . . . . . . 13 10 1.2.3 The Superposition Principle 16 1.3 Solving systems of two coupled 1st order ODE's. 16 1.3.1 The system in matrix form. 16 1.3.2 The Homogeneous case with constant coefficients. 17 1.4 Theory and Theorems for first order systems. 30 1.5 Homogeneous Constant Coefficient Linear 2-dimensional Systems and the Phase Plane 32 1.5.1 The Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.2 Phase Portraits for Real Eigenvalues and Direction Fields. 36 1.5.3 Phase Portraits for Complex Eigenvalues . . . . . . . . . . . . . . 47 1.5.4 SUMMARY Of 6 Types of LINEAR PHASE PORTRAITS in 2D . 50 1.6 Critical Points and Stability 52 1.6.1 Critical Points 1.6.2 Stability of Critical points. 1.7 Non homogeneous Linear systems . 1.8 Nonlinear Systems 58 1.8.1 Solving for the Phase Curves . 52 53 57 58 1.8.2 Critical Points for Nonlinear Systems. 61
CONTENTS 3 1.8.3 Linearization of Nonlinear Systems. 1.9 Diagonalization and 2D Phase Portraits 1.9.1 Relevance to 2D Phase Portraits. 62 1.8.4 Summary for Nonlinear Systems 70 71 71 2 Laplace Transforms 73 2.1 Finding the Laplace Transform of a function . 2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 . 73 2.1.2 The Laplace Transform of simple functions. . . . . . . . . . . . . . 73 2.1.3 The Laplace Transform for Piecewise Continuous functions. 2.1.4 The First Shifting Theorem. 2.1.5 Summary of Laplace Transforms . 2.1.6 Inverting Laplace Transforms. 81 2.1.7 The Gamma Function and L(ta), where a is not an integer. 83 2.2 Laplace Transforms of Derivatives and Solving Simple Linear Constant Coefficient ODEs and Systems of ODE's. 84 2.2.1 The Laplace Transform of the differential of a function. 77 79 80 . . . . . . . . 84 2.2.2 Solving Linear ODEs . . . 2.2.3 Forcing Functions and Transfer Functions. . . . . . . . . . . . . . .