| Introduction | | 1 | |
| | 1 | |
| Conventions Used in This Book |
| | 1 | |
| | 2 | |
| | 2 | |
| How This Book is Organized |
| | 2 | |
| Focusing on First Order Differential Equations |
| | 3 | |
| Surveying Second and Higher Order Differential Equations |
| | 3 | |
| The Power Stuff: Advanced Techniques |
| | 3 | |
| | 3 | |
| | 4 | |
| | 4 | |
| Part I: Focusing on First Order Differential Equations |
| | 5 | |
| Welcome to the World of Differential Equations |
| | 7 | |
| The Essence of Differential Equations |
| | 8 | |
| Derivatives: The Foundation of Differential Equations |
| | 11 | |
| Derivatives that are constants |
| | 11 | |
| Derivatives that are powers |
| | 12 | |
| Derivatives involving trigonometry |
| | 12 | |
| Derivatives involving multiple functions |
| | 12 | |
| Seeing the Big Picture with Direction Fields |
| | 13 | |
| Plotting a direction field |
| | 13 | |
| Connecting slopes into an integral curve |
| | 14 | |
| Recognizing the equilibrium value |
| | 16 | |
| Classifying Differential Equations |
| | 17 | |
| Classifying equations by order |
| | 17 | |
| Classifying ordinary versus partial equations |
| | 17 | |
| Classifying linear versus nonlinear equations |
| | 18 | |
| Solving First Order Differential Equations |
| | 19 | |
| Tackling Second Order and Higher Order Differential Equations |
| | 20 | |
| Having Fun with Advanced Techniques |
| | 21 | |
| Looking at Linear First Order Differential Equations |
| | 23 | |
| First Things First: The Basics of Solving Linear First Order Differential Equations |
| | 24 | |
| Applying initial conditions from the start |
| | 24 | |
| Stepping up to solving differential equations involving functions |
| | 25 | |
| Adding a couple of constants to the mix |
| | 26 | |
| Solving Linear First Order Differential Equations with Integrating Factors |
| | 26 | |
| Solving for an integrating factor |
| | 27 | |
| Using an integrating factor to solve a differential equation |
| | 28 | |
| Moving on up: Using integrating factors in differential equations with functions |
| | 29 | |
| Trying a special shortcut |
| | 30 | |
| Solving an advanced example |
| | 32 | |
| Determining Whether a Solution for a Linear First Order Equation Exists |
| | 35 | |
| Spelling out the existence and uniqueness theorem for linear differential equations |
| | 35 | |
| Finding the general solution |
| | 36 | |
| Checking out some existence and uniqueness examples |
| | 37 | |
| Figuring OUt Whether a Solution for a Nonlinear Differential Equation Exists |
| | 38 | |
| The existence and uniqueness theorem for nonlinear differential equations |
| | 39 | |
| A couple of nonlinear existence and uniqueness examples |
| | 39 | |
| Sorting Out Separable First Order Differential Equations |
| | 41 | |
| Beginning with the Basics of Separable Differential Equations |
| | 42 | |
| Starting easy: Linear separable equations |
| | 43 | |
| Introducing implicit solutions |
| | 43 | |
| Finding explicit solutions from implicit solutions |
| | 45 | |
| Tough to crack: When you can't find an explicit solution |
| | 48 | |
| A neat trick: Turning nonlinear separable equations into linear separable equations |
| | 49 | |
| Trying Out Some Real World Separable Equations |
| | 52 | |
| Getting in control with a sample flow problem |
| | 52 | |
| Striking it rich with a sample monetary problem |
| | 55 | |
| Break It Up! Using Partial Fractions in Separable Equations |
| | 59 | |
| Exploring Exact First Order Differential Equations and Euler's Method |
| | 63 | |
| Exploring the Basics of Exact Differential Equations |
| | 63 | |
| Defining exact differential equations |
| | 64 | |
| Working out a typical exact differential equation |
| | 65 | |
| Determining Whether a Differential Equation Is Exact |
| | 66 | |
| Checking out a useful theorem |
| | 66 | |
| | 67 | |
| Conquering Nonexact Differential Equations with Integrating Factors |
| | 70 | |
| Finding an integrating factor |
| | 71 | |
| Using an integrating factor to get an exact equation |
| | 73 | |
| The finishing touch: Solving the exact equation |
| | 74 | |
| Getting Numerical with Euler's Method |
| | 75 | |
| | 76 | |
| Checking the method's accuracy on a computer |
| | 77 | |
| Delving into Difference Equations |
| | 83 | |
| | 84 | |
| | 84 | |
| | 85 | |
| Part II: Surveying Second and Higher Order Differential Equations |
| | 89 | |
| Examining Second Order Linear Homogeneous Differential Equations |
| | 91 | |
| The Basics of Second Order Differential Equations |
| | 91 | |
| | 92 | |
| | 93 | |
| Second Order Linear Homogeneous Equations with Constant Coefficients |
| | 94 | |
| | 94 | |
| | 95 | |
| Checking Out Characteristic Equations |
| | 96 | |
| | 97 | |
| | 100 | |
| | 106 | |
| Getting a Second Solution by Reduction of Order |
| | 109 | |
| Seeing how reduction of order works |
| | 110 | |
| | 111 | |
| Putting Everything Together with Some Handy Theorems |
| | 114 | |
| | 114 | |
| | 115 | |
| | 117 | |
| Studying Second Order Linear Nonhomogeneous Differential Equations |
| | 123 | |
| The General Solution of Second Order Linear Nonhomogeneous Equations |
| | 124 | |
| Understanding an important theorem |
| | 124 | |
| Putting the theorem to work |
| | 125 | |
| Finding Particular Solutions with the Method of Undetermined Coefficients |
| | 127 | |
| When g(x) is in the form of erx |
| | 127 | |
| When g(x) is a polynomial of order n |
| | 128 | |
| When g(x) is a combination of sines and cosines |
| | 131 | |
| When g(x) is a Product of two different forms |
| | 133 | |
| Breaking Down Equations with the Variation of Parameters Method |
| | 135 | |
| Nailing down the basics of the method |
| | 136 | |
| Solving a typical example |
| | 137 | |
| Applying the method to any linear equation |
| | 138 | |
| What a Pair! The Variation of parameters method meets the Wronskian |
| | 142 | |
| Bouncing Around with Springs `n' things |
| | 143 | |
| | 144 | |
| | 148 | |
| Handling Higher Order Linear Homogeneous Differential Equations |
| | 151 | |
| The Write Stuff: The Notation of Higher Order Differential Equations |
| | 152 | |
| Introducing the Basics of Higher Order Linear Homogeneous Equations |
| | 153 | |
| The format, solutions, and initial conditions |
| | 153 | |
| A couple of cool theorems |
| | 155 | |
| Tackling Different Types of Higher Order Linear Homogeneous Equations |
| | 156 | |
| | 156 | |
| | 161 | |
| | 164 | |
| | 166 | |
| Taking On Higher Order Linear Nonhomogeneous Differential Equations |
| | 173 | |
| Mastering the Method of Undetermined Coefficients for Higher Order Equations |
| | 174 | |
| When g(x) is in the form erx |
| | 176 | |
| When g(x) is a polynomial of order n |
| | 179 | |
| When g(x) is a combination of sines and cosines |
| | 182 | |
| Solving Higher Order Equations with Variation of Parameters |
| | 185 | |
| | 185 | |
| Working through an example |
| | 186 | |
| Part III: The Power Stuff: Advanced Techniques |
| | 189 | |
| Getting Serious with Power Series and Ordinary Points |
| | 191 | |
| Perusing the Basics of Power Series |
| | 191 | |
| Determining Whether a Power Series Converges with the Ratio Test |
| | 192 | |
| The fundamentals of the ratio test |
| | 192 | |
| | 193 | |
| Shifting the Series Index |
| | 195 | |
| Taking a Look at the Taylor Series |
| | 195 | |
| Solving Second Order Differential Equations with Power Series |
| | 196 | |
| When you already know the solution |
| | 198 | |
| When you don't know the solution beforehand |
| | 204 | |
| A famous problem: Airy's equation |
| | 207 | |
| Powering through Singular Points |
| | 213 | |
| Pointing Out the Basics of Singular Points |
| | 213 | |
| | 214 | |
| The behavior of singular points |
| | 214 | |
| Regular versus irregular singular points |
| | 215 | |
| Exploring Exciting Euler Equations |
| | 219 | |
| | 220 | |
| | 222 | |
| | 223 | |
| Putting it all together with a theorem |
| | 224 | |
| Figuring Series Solutions Near Regular Singular Points |
| | 225 | |
| Identifying the general solutions |
| | 225 | |
| The basics of solving equations near singular points |
| | 227 | |
| A numerical example of solving an equation near singular points |
| | 230 | |
| Taking a closer look at indicial equations |
| | 235 | |
| Working with Laplace Transforms |
| | 239 | |
| Breaking Down a Typical Laplace Transform |
| | 239 | |
| Deciding Whether a Laplace Transform Converges |
| | 240 | |
| Calculating Basic Laplace Transforms |
| | 241 | |
| | 242 | |
| | 242 | |
| | 242 | |
| Consulting a handy table for some relief |
| | 244 | |
| Solving Differential Equations with Laplace Transforms |
| | 245 | |
| A few theorems to send you on your way |
| | 246 | |
| Solving a second order homogeneous equation |
| | 247 | |
| Solving a second order nonhomogeneous equation |
| | 251 | |
| Solving a higher order equation |
| | 255 | |
| Factoring Laplace Transforms and Convolution Integrals |
| | 258 | |
| Factoring a Laplace transform into fractions |
| | 258 | |
| Checking out convolution integrals |
| | 259 | |
| | 261 | |
| Defining the step function |
| | 261 | |
| Figuring the Laplace transform of the step function |
| | 262 | |
| Tackling Systems of First Order Linear Differential Equations |
| | 265 | |
| Introducing the Basics of Matrices |
| | 266 | |
| | 266 | |
| Working through the algebra |
| | 267 | |
| | 268 | |
| Mastering Matrix Operations |
| | 269 | |
| | 269 | |
| | 270 | |
| | 270 | |
| Multiplication of a matrix and a number |
| | 270 | |
| Multiplication of two matrices |
| | 270 | |
| Multiplication of a matrix and a vector |
| | 271 | |
| | 272 | |
| | 272 | |
| Having Fun with Eigenvectors `n' Things |
| | 278 | |
| | 278 | |
| Eigenvalues and eigenvectors |
| | 281 | |
| Solving Systems of First-Order Linear Homogeneous Differential Equations |
| | 283 | |
| | 284 | |
| Making your way through an example |
| | 285 | |
| solving systems of First Order Linear Nonhomogeneous Equations |
| | 288 | |
| Assuming the correct form of the particular solution |
| | 289 | |
| | 290 | |
| | 292 | |
| Discovering Three Fail-Proof Numerical Methods |
| | 293 | |
| Number Crunching with Euler's Method |
| | 294 | |
| The fundamentals of the method |
| | 294 | |
| Using code to see the method in action |
| | 295 | |
| Moving On Up with the Improved Euler's Method |
| | 299 | |
| Understanding the improvements |
| | 300 | |
| | 300 | |
| Plugging a steep slope into the new code |
| | 304 | |
| Adding Even More Precision with the Runge-Kutta Method |
| | 308 | |
| The method's recurrence relation |
| | 308 | |
| Working with the method in code |
| | 309 | |
| Part IV: The Part of Tens |
| | 315 | |
| Ten Super-Helpful Online Differential Equation Tutorials |
| | 317 | |
| AnalyzeMath.com's Introduction to Differential Equations |
| | 317 | |
| Harvey Mudd College Mathematics Online Tutorial |
| | 318 | |
| John Appleby's Introduction to Differential Equations |
| | 318 | |
| | 318 | |
| Martin J. Osborne's Differential Equation Tutorial |
| | 318 | |
| Midnight Tutor's Video Tutorial |
| | 319 | |
| The Ohio State University Physics Department's Introduction to Differential Equations |
| | 319 | |
| | 319 | |
| | 319 | |
| University of Surrey Tutorial |
| | 320 | |
| Ten Really Cool Online Differential Equation Solving Tools |
| | 321 | |
| AnalyzeMath.com's Runge-Kutta Method Applet |
| | 321 | |
| Coolmath.com's Graphing Calculator |
| | 321 | |
| | 322 | |
| An Equation Solver from QuickMath Automatic Math Solutions |
| | 322 | |
| First Order Differential Equation Solver |
| | 322 | |
| GCalc Online Graphing Calculator |
| | 322 | |
| | 323 | |
| Math @ CowPi's System Solver |
| | 323 | |
| A Matrix Inverter from QuickMath Automatic Math Solutions |
| | 323 | |
| Visual Differential Equation Solving Applet |
| | 323 | |
| Index | | 325 | |
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