Differential Equations and Linear Algebra
                              Lecture Notes
                             Simon J.A. Malham
                  Department of Mathematics, Heriot-Watt University
                                                           Contents
                           Chapter 1.    Linear second order ODEs                                     5
                              1.1.  Newton’s second law                                               5
                              1.2.  Springs and Hooke’s Law                                           6
                              1.3.  General ODEs and their classification                             10
                              1.4.  Exercises                                                        12
                           Chapter 2.    Homogeneous linear ODEs                                     15
                              2.1.  The Principle of Superposition                                   15
                              2.2.  Linear second order constant coefficient homogeneous ODEs          15
                              2.3.  Practical example: damped springs                                20
                              2.4.  Exercises                                                        22
                           Chapter 3.    Non-homogeneous linear ODEs                                 23
                              3.1.  Example applications                                             23
                              3.2.  Linear operators                                                 24
                              3.3.  Solving non-homogeneous linear ODEs                              25
                              3.4.  Method of undetermined coefficients                                26
                              3.5.  Initial and boundary value problems                              28
                              3.6.  Degenerate inhomogeneities                                       30
                              3.7.  Resonance                                                        33
                              3.8.  Equidimensional equations                                        37
                              3.9.  Exercises                                                        38
                              Summary: solving linear constant coefficient second order IVPs           40
                           Chapter 4.    Laplace transforms                                          41
                              4.1.  Introduction                                                     41
                              4.2.  Properties of Laplace transforms                                 43
                              4.3.  Solving linear constant coefficients ODEs via Laplace transforms   44
                              4.4.  Impulses and Dirac’s delta function                              46
                              4.5.  Exercises                                                        50
                              Table of Laplace transforms                                            52
                           Chapter 5.    Linear algebraic equations                                  53
                              5.1.  Physical and engineering applications                            53
                              5.2.  Systems of linear algebraic equations                            54
                              5.3.  Gaussian elimination                                             57
                              5.4.  Solution of general rectangular systems                          63
                                                                 3
                               4                                   CONTENTS
                                 5.5.   Matrix Equations                                                         63
                                 5.6.   Linear independence                                                      66
                                 5.7.   Rank of a matrix                                                         68
                                 5.8.   Fundamental theorem for linear systems                                   69
                                 5.9.   Gauss-Jordan method                                                      70
                                 5.10.   Matrix Inversion via EROs                                               71
                                 5.11.   Exercises                                                               73
                               Chapter 6.    Linear algebraic eigenvalue problems                                75
                                 6.1.   Eigenvalues and eigenvectors                                             75
                                 6.2.   Diagonalization                                                          82
                                 6.3.   Exercises                                                                83
                               Chapter 7.    Systems of differential equations                                    85
                                 7.1.   Linear second order systems                                              85
                                 7.2.   Linear second order scalar ODEs                                          88
                                 7.3.   Higher order linear ODEs                                                 90
                                 7.4.   Solution to linear constant coefficient ODE systems                        90
                                 7.5.   Solution to general linear ODE systems                                   92
                                 7.6.   Exercises                                                                92
                               Bibliography                                                                      95