First Order Equations
We start our study of differential equations in the same way the pioneers in this field did.
We show particular techniques to solve particular types of first order differential equations.
The techniques were developed in the eighteenth and nineteenth centuries and the equat...
ordinary differential equations gabriel nagy mathematics department
michigan state university
48824 gnagymsuedu january 18
2021 x2 x1 x 1 x 2 b a 0 contents preface 1 chapter
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Ordinary Differential Equations Gabriel Nagy
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Ordinary Differential Equations
Gabriel Nagy
Mathematics Department,
Michigan State University,
East Lansing, MI, 48824.
gnagy@msu.edu
January 18, 2021
x2
x2
x1
b a
0 x1
,
,
, Contents
Preface 1
Chapter 1. First Order Equations 3
1.1. Linear Constant Coefficient Equations 4
1.1.1. Overview of Differential Equations 4
1.1.2. Linear Differential Equations 5
1.1.3. Solving Linear Differential Equations 6
1.1.4. The Integrating Factor Method 8
1.1.5. The Initial Value Problem 10
1.1.6. Exercises 13
1.2. Linear Variable Coefficient Equations 14
1.2.1. Review: Constant Coefficient Equations 14
1.2.2. Solving Variable Coefficient Equations 15
1.2.3. The Initial Value Problem 17
1.2.4. The Bernoulli Equation 19
1.2.5. Exercises 23
1.3. Separable Equations 24
1.3.1. Separable Equations 24
1.3.2. Euler Homogeneous Equations 29
1.3.3. Solving Euler Homogeneous Equations 32
1.3.4. Exercises 35
1.4. Exact Differential Equations 36
1.4.1. Exact Equations 36
1.4.2. Solving Exact Equations 37
1.4.3. Semi-Exact Equations 41
1.4.4. The Equation for the Inverse Function 46
1.4.5. Exercises 50
1.5. Applications of Linear Equations 51
1.5.1. Exponential Decay 51
1.5.2. Carbon-14 Dating 52
1.5.3. Newton’s Cooling Law 53
1.5.4. Mixing Problems 54
1.5.5. Exercises 59
1.6. Nonlinear Equations 60
1.6.1. The Picard-Lindelöf Theorem 60
1.6.2. Comparison of Linear and Nonlinear Equations 69
1.6.3. Direction Fields 71
1.6.4. Exercises 75
Chapter 2. Second Order Linear Equations 77
2.1. Variable Coefficients 78
III
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