Differential Equations I
MATB44H3F
Version September 15, 2011-1949
,ii
,Contents
1 Introduction 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sample Application of Differential Equations . . . . . . . . . . . 2
2 First Order Ordinary Differential Equations 5
2.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Exact Differential Equations . . . . . . . . . . . . . . . . . . . . . 7
2.3 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Linear First Order Equations . . . . . . . . . . . . . . . . . . . . 14
2.5 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . 17
2.5.2 Homogeneous Equations . . . . . . . . . . . . . . . . . . . 19
2.5.3 Substitution to Reduce Second Order Equations to First
Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Applications and Examples of First Order ode’s 25
3.1 Orthogonal Trajectories . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . 27
3.3 Population Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Predator-Prey Models . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Newton’s Law of Cooling . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Water Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Motion of Objects Falling Under Gravity with Air Resistance . . 34
3.8 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Particle Moving on a Curve . . . . . . . . . . . . . . . . . . . . . 39
iii
, iv CONTENTS
4 Linear Differential Equations 45
4.1 Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . 47
4.1.1 Linear Differential Equations with Constant Coefficients . 52
4.2 Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . 54
5 Second Order Linear Equations 57
5.1 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 60
5.2.1 Shortcuts for Undetermined Coefficients . . . . . . . . . . 64
5.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 66
6 Applications of Second Order Differential Equations 71
6.1 Motion of Object Hanging from a Spring . . . . . . . . . . . . . . 71
6.2 Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7 Higher Order Linear Differential Equations 79
7.1 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Substitutions: Euler’s Equation . . . . . . . . . . . . . . . . . . . 82
8 Power Series Solutions to Linear Differential Equations 85
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.2 Background Knowledge Concerning Power Series . . . . . . . . . 88
8.3 Analytic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.4 Power Series Solutions: Levels of Success . . . . . . . . . . . . . . 91
8.5 Level 1: Finding a finite number of coefficients . . . . . . . . . . 91
8.6 Level 2: Finding the recursion relation . . . . . . . . . . . . . . . 94
8.7 Solutions Near a Singular Point . . . . . . . . . . . . . . . . . . . 97
8.8 Functions Defined via Differential Equations . . . . . . . . . . . . 111
8.8.1 Chebyshev Equation . . . . . . . . . . . . . . . . . . . . . 111
8.8.2 Legendre Equation . . . . . . . . . . . . . . . . . . . . . . 113
8.8.3 Airy Equation . . . . . . . . . . . . . . . . . . . . . . . . 115
8.8.4 Laguerre’s Equation . . . . . . . . . . . . . . . . . . . . . 115
8.8.5 Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Linear Systems 121
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2 Computing eT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.3 The 2 × 2 Case in Detail . . . . . . . . . . . . . . . . . . . . . . . 129
9.4 The Non-Homogeneous Case . . . . . . . . . . . . . . . . . . . . 133
MATB44H3F
Version September 15, 2011-1949
,ii
,Contents
1 Introduction 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sample Application of Differential Equations . . . . . . . . . . . 2
2 First Order Ordinary Differential Equations 5
2.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Exact Differential Equations . . . . . . . . . . . . . . . . . . . . . 7
2.3 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Linear First Order Equations . . . . . . . . . . . . . . . . . . . . 14
2.5 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . 17
2.5.2 Homogeneous Equations . . . . . . . . . . . . . . . . . . . 19
2.5.3 Substitution to Reduce Second Order Equations to First
Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Applications and Examples of First Order ode’s 25
3.1 Orthogonal Trajectories . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . 27
3.3 Population Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Predator-Prey Models . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Newton’s Law of Cooling . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Water Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Motion of Objects Falling Under Gravity with Air Resistance . . 34
3.8 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Particle Moving on a Curve . . . . . . . . . . . . . . . . . . . . . 39
iii
, iv CONTENTS
4 Linear Differential Equations 45
4.1 Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . 47
4.1.1 Linear Differential Equations with Constant Coefficients . 52
4.2 Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . 54
5 Second Order Linear Equations 57
5.1 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 60
5.2.1 Shortcuts for Undetermined Coefficients . . . . . . . . . . 64
5.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 66
6 Applications of Second Order Differential Equations 71
6.1 Motion of Object Hanging from a Spring . . . . . . . . . . . . . . 71
6.2 Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7 Higher Order Linear Differential Equations 79
7.1 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Substitutions: Euler’s Equation . . . . . . . . . . . . . . . . . . . 82
8 Power Series Solutions to Linear Differential Equations 85
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.2 Background Knowledge Concerning Power Series . . . . . . . . . 88
8.3 Analytic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.4 Power Series Solutions: Levels of Success . . . . . . . . . . . . . . 91
8.5 Level 1: Finding a finite number of coefficients . . . . . . . . . . 91
8.6 Level 2: Finding the recursion relation . . . . . . . . . . . . . . . 94
8.7 Solutions Near a Singular Point . . . . . . . . . . . . . . . . . . . 97
8.8 Functions Defined via Differential Equations . . . . . . . . . . . . 111
8.8.1 Chebyshev Equation . . . . . . . . . . . . . . . . . . . . . 111
8.8.2 Legendre Equation . . . . . . . . . . . . . . . . . . . . . . 113
8.8.3 Airy Equation . . . . . . . . . . . . . . . . . . . . . . . . 115
8.8.4 Laguerre’s Equation . . . . . . . . . . . . . . . . . . . . . 115
8.8.5 Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Linear Systems 121
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2 Computing eT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.3 The 2 × 2 Case in Detail . . . . . . . . . . . . . . . . . . . . . . . 129
9.4 The Non-Homogeneous Case . . . . . . . . . . . . . . . . . . . . 133
