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
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
