Fundamentals of ODE

153 Pages·1997·0.524 MB·English

by  Xu, Labute.

FUNDAMENTALS OF ORDINARY DIFFER-
ENTIAL EQUATIONS — THE LECTURE
NOTES FOR COURSE (189) 261/325

FUNDAMENTALS OF ORDINARY DIFFER-
ENTIAL EQUATIONS
JIAN-JUN XU AND JOHN LABUTE
DepartmentofMathematicsandStatistics,McGillUniversity
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 Linear Equation: 2
1.5 Homogeneous Linear Equation: 2
1.6 Partial Differential Equation (PDE) 2
1.7 General Solution of a Linear Differential Equation 3
1.8 A System of ODE’s 3
2 The Approaches of Finding Solutions of ODE 4
2.1 Analytical Approaches 4
2.2 Numerical Approaches 4
2. FIRST ORDER DIFFERENTIAL EQUATIONS 5
1 Linear Equation 7
1.1 Linear homogeneous equation 7
1.2 Linear inhomogeneous equation 8
2 Separable Equations. 11
3 Logistic Equation 13
4 Fundamental Existence and Uniqueness Theorem 14
5 Bernoulli Equation: 15
6 Homogeneous Equation: 16
7 Exact Equations. 19
8 Theorem. 20
9 Integrating Factors. 21
v
vi FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS
10 Change of Variables. 23
10.1 y(cid:48) = f(ax+by), b (cid:54)= 0 23
dy a x+b y+c
1 1 1
10.2 = 23
dx a x+b y+c
2 2 2
10.3 Riccatti equation: y(cid:48) = p(x)y+q(x)y2+r(x) 24
11 Orthogonal Trajectories. 25
12 Falling Bodies with Air Resistance 27
13 Mixing Problems 27
14 Heating and Cooling Problems 28
15 Radioactive Decay 29
16 Definitions and Basic Concepts 31
16.1 Directional Field 31
16.2 Integral Curves 31
16.3 Autonomous Systems 31
16.4 Equilibrium Points 31
17 Phase Line Analysis 32
18 Bifurcation Diagram 32
19 Euler’s Method 37
20 Improved Euler’s Method 38
21 Higher Order Methods 38
3. N-TH ORDER DIFFERENTIAL EQUATIONS 43
1 Theorem of Existence and Uniqueness (I) 46
1.1 Lemma 46
2 Theorem of Existence and Uniqueness (II) 47
3 Theorem of Existence and Uniqueness (III) 47
3.1 Case (I) 49
3.2 Case (II) 50
4 Linear Equations 50
4.1 Basic Concepts and General Properties 50
5 Basic Theory of Linear Differential Equations 51
5.1 Basics of Linear Vector Space 51
5.1.1 Isomorphic Linear Transformation 51
5.1.2 Dimension and Basis of Vector Space 52
5.1.3 (*) Span and Subspace 52
5.1.4 Linear Independency 52
5.2 Wronskian of n-functions 53
Contents vii
5.2.1 Definition 53
5.2.2 Theorem 1 54
5.2.3 Theorem 2 54
6 The Method with Undetermined Parameters 57
6.1 Basic Equalities (I) 57
6.2 Cases (I) ( r > r ) 58
1 2
6.3 Cases (II) ( r = r ) 59
1 2
6.4 Cases (III) ( r = λ±iµ) 60
1,2
7 The Method with Differential Operator 61
7.1 Basic Equalities (II). 61
7.2 Cases (I) ( b2−4ac > 0) 62
7.3 Cases (II) ( b2−4ac = 0) 62
7.4 Cases (III) ( b2−4ac < 0) 63
7.5 Theorems 64
8 The Differential Operator for Equations with Constant
Coefficients 67
9 The Method of Variation of Parameters 68
10 Euler Equations 71
11 Exact Equations 73
12 Reduction of Order 74
13 (*) Vibration System 77
4. SERIES SOLUTION OF LINEAR
DIFFERENTIAL EQUATIONS 81
1 Series Solutions near a Ordinary Point 84
1.1 Theorem 84
2 Series Solutions near a Regular Singular Point 87
2.1 Case (I): The roots (r −r (cid:54)= N) 89
1 2
2.2 Case (II): The roots (r = r ) 89
1 2
2.3 Case (III): The roots (r −r = N > 0) 90
1 2
3 Bessel Equation 95
4 The Case of Non-integer ν 96
5 The Case of ν = −m with m an integer ≥ 0 96
5. LAPLACE TRANSFORMS 101
1 Introduction 103
2 Laplace Transform 104
2.1 Definition 104
viii FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS
2.2 Basic Properties and Formulas 105
2.2.1 Linearity of the transform 105
2.2.2 Formula (I) 106
2.2.3 Formula (II) 106
2.2.4 Formula (III) 106
3 Inverse Laplace Transform 107
3.1 Theorem: 107
3.2 Definition 107
4 Solve IVP of DE’s with Laplace Transform Method 109
4.1 Example 1 109
4.2 Example 2 111
5 Step Function 113
5.1 Definition 113
5.2 Laplace transform of unit step function 113
6 Impulse Function 113
6.1 Definition 113
6.2 Laplace transform of unit step function 114
7 Convolution Integral 114
7.1 Theorem 114
6. (*) SYSTEMS OF LINEAR DIFFERENTIAL
EQUATIONS 115
1 Mathematical Formulation of a Practical Problem 117
2 (2×2) System of Linear Equations 119
2.1 Case 1: ∆ > 0 119
2.2 Case 2: ∆ < 0 120
2.3 Case 3: ∆ = 0 121
Appendices 127
ASSIGNMENTS AND SOLUTIONS 127

Chapter 1
INTRODUCTION
1. Definitions and Basic Concepts
1.1 Ordinary Differential Equation (ODE)
An equation involving the derivatives of an unknown function y of a
single variable x over an interval x ∈ (I).
1.2 Solution
Any function y = f(x) which satisfies this equation over the interval
(I) is called a solution of the ODE.
For example, y = e2x is a solution of the ODE
y(cid:48) = 2y
and y = sin(x2) is a solution of the ODE
xy(cid:48)(cid:48)−y(cid:48)+4x3y = 0.
1.3 Order n of the DE
An ODE is said to be order n, if y(n) is the highest order derivative
occurring in the equation. The simplest first order ODE is y(cid:48) = g(x).
The most general form of an n-th order ODE is
F(x,y,y(cid:48),...,y(n)) = 0
with F a function of n+2 variables x,u ,u ,...,u . The equations
0 1 n
xy(cid:48)(cid:48)+y = x3, y(cid:48)+y2 = 0, y(cid:48)(cid:48)(cid:48)+2y(cid:48)+y = 0
1