test bank for ordinary differential equations with all questions and answers made farmiiar and a 100% guarantee pass .new edition

Mathematics Department,




Ordinary Differential Equations




Gabriel Nagy


Mathematics Department,
Michigan State University,
East Lansing, MI, 48824.


January 18, 2021



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, Ordinary Differential Equations
Accreditted 100% pass.




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




1

,IV CONTENTS


2.1.1. Definitions and Examples 78
2.1.2. Solutions to the Initial Value Problem. 80
2.1.3. Properties of Homogeneous Equations 81
2.1.4. The Wronskian Function 85
2.1.5. Abel’s Theorem 86
2.1.6. Exercises 89
2.2. Reduction of Order Methods 90
2.2.1. Special Second Order Equations 90
2.2.2. Conservation of the Energy 93
2.2.3. The Reduction of Order Method 98
2.2.4. Exercises 101
2.3. Homogenous Constant Coefficients Equations 102
2.3.1. The Roots of the Characteristic Polynomial 102
2.3.2. Real Solutions for Complex Roots 106
2.3.3. Constructive Proof of Theorem 2.3.2 108
2.3.4. Exercises 111
2.4. Euler Equidimensional Equation 112
2.4.1. The Roots of the Indicial Polynomial 112
2.4.2. Real Solutions for Complex Roots 115
2.4.3. Transformation to Constant Coefficients 117
2.4.4. Exercises 119
2.5. Nonhomogeneous Equations 120
2.5.1. The General Solution Formula 120
2.5.2. The Undetermined Coefficients Method 121
2.5.3. The Variation of Parameters Method 125
2.5.4. Exercises 130
2.6. Applications 131
2.6.1. Review of Constant Coefficient Equations 131
2.6.2. Undamped Mechanical Oscillations 132
2.6.3. Damped Mechanical Oscillations 134
2.6.4. Electrical Oscillations 136
2.6.5. Exercises 139

Chapter 3. Power Series Solutions 141
3.1. Solutions Near Regular Points 143
3.1.1. Regular Points 143
3.1.2. The Power Series Method 144
3.1.3. The Legendre Equation 151
3.1.4. Exercises 155
3.2. Solutions Near Regular Singular Points 156
3.2.1. Regular Singular Points 156
3.2.2. The Frobenius Method 159
3.2.3. The Bessel Equation 163
3.2.4. Exercises 168
Notes on Chapter 3 169

Chapter 4. The Laplace Transform Method 173
4.1. Introduction to the Laplace Transform 175
4.1.1. Oveview of the Method 175
4.1.2. The Laplace Transform 176

, CONTENTS V


4.1.3. Main Properties 180
4.1.4. Solving Differential Equations 184
4.1.5. Exercises 186
4.2. The Initial Value Problem 187
4.2.1. Solving Differential Equations 187
4.2.2. One-to-One Property 188
4.2.3. Partial Fractions 190
4.2.4. Higher Order IVP 195
4.2.5. Exercises 197
4.3. Discontinuous Sources 198
4.3.1. Step Functions 198
4.3.2. The Laplace Transform of Steps 199
4.3.3. Translation Identities 200
4.3.4. Solving Differential Equations 204
4.3.5. Exercises 209
4.4. Generalized Sources 210
4.4.1. Sequence of Functions and the Dirac Delta 210
4.4.2. Computations with the Dirac Delta 212
4.4.3. Applications of the Dirac Delta 214
4.4.4. The Impulse Response Function 215
4.4.5. Comments on Generalized Sources 218
4.4.6. Exercises 221
4.5. Convolutions and Solutions 222
4.5.1. Definition and Properties 222
4.5.2. The Laplace Transform 224
4.5.3. Solution Decomposition 226
4.5.4. Exercises 230

Chapter 5. Systems of Linear Differential Equations 231
5.1. General Properties 232
5.1.1. First Order Linear Systems 232
5.1.2. Existence of Solutions 234
5.1.3. Order Transformations 235
5.1.4. Homogeneous Systems 238
5.1.5. The Wronskian and Abel’s Theorem 242
5.1.6. Exercises 246
5.2. Solution Formulas 247
5.2.1. Homogeneous Systems 247
5.2.2. Homogeneous Diagonalizable Systems 249
5.2.3. Nonhomogeneous Systems 256
5.2.4. Exercises 259
5.3. Two-Dimensional Homogeneous Systems 260
5.3.1. Diagonalizable Systems 260
5.3.2. Non-Diagonalizable Systems 263
5.3.3. Exercises 266
5.4. Two-Dimensional Phase Portraits 267
5.4.1. Real Distinct Eigenvalues 268
5.4.2. Complex Eigenvalues 271
5.4.3. Repeated Eigenvalues 273
5.4.4. Exercises 275




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