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Table of Contents
Introduction..................................................................1
About This Book...............................................................................................1
Conventions Used in This Book .....................................................................1
What You’re Not to Read.................................................................................2
Foolish Assumptions .......................................................................................2
How This Book Is Organized...........................................................................2
Part I: Focusing on First Order Differential Equations.......................3
Part II: Surveying Second and Higher Order
Differential Equations.........................................................................3
Part III: The Power Stuff: Advanced Techniques ................................3
Part IV: The Part of Tens........................................................................3
Icons Used in This Book..................................................................................4
Where to Go from Here....................................................................................4
Part I: Focusing on First Order Differential Equations ......5
Chapter 1: 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
Chapter 2: 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
Chapter 3: 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
xii Differential Equations For Dummies
Chapter 4: 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
Applying the theorem..........................................................................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
Understanding the method .................................................................76
Checking the method’s accuracy on a computer.............................77
Delving into Difference Equations................................................................83
Some handy terminology ....................................................................84
Iterative solutions ................................................................................84
Equilibrium solutions ..........................................................................85
Part II: Surveying Second and Higher Order
Differential Equations..................................................89
Chapter 5: Examining Second Order Linear
Homogeneous Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .91
The Basics of Second Order Differential Equations...................................91
Linear equations...................................................................................92
Homogeneous equations.....................................................................93
Second Order Linear Homogeneous Equations
with Constant Coefficients ........................................................................94
Elementary solutions ...........................................................................94
Initial conditions...................................................................................95
Checking Out Characteristic Equations ......................................................96
Real and distinct roots.........................................................................97
Complex roots.....................................................................................100
Identical real roots .............................................................................106
Getting a Second Solution by Reduction of Order...................................109
Seeing how reduction of order works..............................................110
Trying out an example.......................................................................111
Table of Contents xiii
Putting Everything Together with Some Handy Theorems ....................114
Superposition......................................................................................114
Linear independence .........................................................................115
The Wronskian....................................................................................117
Chapter 6: 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
A mass without friction .....................................................................144
A mass with drag force ......................................................................148
Chapter 7: 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
Real and distinct roots.......................................................................156
Real and imaginary roots ..................................................................161
Complex roots.....................................................................................164
Duplicate roots ...................................................................................166
xiv Differential Equations For Dummies
Chapter 8: 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
The basics of the method..................................................................185
Working through an example............................................................186
Part III: The Power Stuff: Advanced Techniques...........189
Chapter 9: 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
Plugging in some numbers ................................................................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
Chapter 10: Powering through Singular Points . . . . . . . . . . . . . . . . . .213
Pointing Out the Basics of Singular Points ...............................................213
Finding singular points ......................................................................214
The behavior of singular points .......................................................214
Regular versus irregular singular points.........................................215
Exploring Exciting Euler Equations ...........................................................219
Real and distinct roots.......................................................................220
Real and equal roots ..........................................................................222
Complex roots.....................................................................................223
Putting it all together with a theorem..............................................224
Figuring Series Solutions Near Regular Singular Points..........................225
Identifying the general solution........................................................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
Table of Contents xv
Chapter 11: Working with Laplace Transforms . . . . . . . . . . . . . . . . . .239
Breaking Down a Typical Laplace Transform...........................................239
Deciding Whether a Laplace Transform Converges ................................240
Calculating Basic Laplace Transforms ......................................................241
The transform of 1..............................................................................242
The transform of eat ............................................................................242
The transform of sin at ......................................................................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
Surveying Step Functions............................................................................261
Defining the step function .................................................................261
Figuring the Laplace transform of the step function .....................262
Chapter 12: Tackling Systems of First Order Linear
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265
Introducing the Basics of Matrices ............................................................266
Setting up a matrix .............................................................................266
Working through the algebra ............................................................267
Examining matrices............................................................................268
Mastering Matrix Operations......................................................................269
Equality................................................................................................269
Addition ...............................................................................................270
Subtraction..........................................................................................270
Multiplication of a matrix and a number.........................................270
Multiplication of two matrices..........................................................270
Multiplication of a matrix and a vector ...........................................271
Identity.................................................................................................272
The inverse of a matrix......................................................................272
Having Fun with Eigenvectors ’n’ Things..................................................278
Linear independence .........................................................................278
Eigenvalues and eigenvectors ..........................................................281
Solving Systems of First-Order Linear Homogeneous
Differential Equations ..............................................................................283
Understanding the basics..................................................................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
Crunching the numbers.....................................................................290
Winding up your work .......................................................................292
xvi Differential Equations For Dummies
Chapter 13: 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
Coming up with new code.................................................................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
Chapter 14: 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
Kardi Teknomo’s Page .................................................................................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
Paul’s Online Math Notes ............................................................................319
S.O.S. Math ....................................................................................................319
University of Surrey Tutorial ......................................................................320
Chapter 15: Ten Really Cool Online Differential
Equation Solving Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .321
AnalyzeMath.com’s Runge-Kutta Method Applet ....................................321
Coolmath.com’s Graphing Calculator .......................................................321
Direction Field Plotter .................................................................................322
An Equation Solver from QuickMath Automatic Math Solutions...........322
First Order Differential Equation Solver....................................................322
GCalc Online Graphing Calculator .............................................................322
JavaView Ode Solver....................................................................................323
Math @ CowPi’s System Solver...................................................................323
A Matrix Inverter from QuickMath Automatic Math Solutions ..............323
Visual Differential Equation Solving Applet..............................................323
Introduction..................................................................1
About This Book...............................................................................................1
Conventions Used in This Book .....................................................................1
What You’re Not to Read.................................................................................2
Foolish Assumptions .......................................................................................2
How This Book Is Organized...........................................................................2
Part I: Focusing on First Order Differential Equations.......................3
Part II: Surveying Second and Higher Order
Differential Equations.........................................................................3
Part III: The Power Stuff: Advanced Techniques ................................3
Part IV: The Part of Tens........................................................................3
Icons Used in This Book..................................................................................4
Where to Go from Here....................................................................................4
Part I: Focusing on First Order Differential Equations ......5
Chapter 1: 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
Chapter 2: 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
Chapter 3: 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
xii Differential Equations For Dummies
Chapter 4: 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
Applying the theorem..........................................................................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
Understanding the method .................................................................76
Checking the method’s accuracy on a computer.............................77
Delving into Difference Equations................................................................83
Some handy terminology ....................................................................84
Iterative solutions ................................................................................84
Equilibrium solutions ..........................................................................85
Part II: Surveying Second and Higher Order
Differential Equations..................................................89
Chapter 5: Examining Second Order Linear
Homogeneous Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .91
The Basics of Second Order Differential Equations...................................91
Linear equations...................................................................................92
Homogeneous equations.....................................................................93
Second Order Linear Homogeneous Equations
with Constant Coefficients ........................................................................94
Elementary solutions ...........................................................................94
Initial conditions...................................................................................95
Checking Out Characteristic Equations ......................................................96
Real and distinct roots.........................................................................97
Complex roots.....................................................................................100
Identical real roots .............................................................................106
Getting a Second Solution by Reduction of Order...................................109
Seeing how reduction of order works..............................................110
Trying out an example.......................................................................111
Table of Contents xiii
Putting Everything Together with Some Handy Theorems ....................114
Superposition......................................................................................114
Linear independence .........................................................................115
The Wronskian....................................................................................117
Chapter 6: 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
A mass without friction .....................................................................144
A mass with drag force ......................................................................148
Chapter 7: 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
Real and distinct roots.......................................................................156
Real and imaginary roots ..................................................................161
Complex roots.....................................................................................164
Duplicate roots ...................................................................................166
xiv Differential Equations For Dummies
Chapter 8: 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
The basics of the method..................................................................185
Working through an example............................................................186
Part III: The Power Stuff: Advanced Techniques...........189
Chapter 9: 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
Plugging in some numbers ................................................................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
Chapter 10: Powering through Singular Points . . . . . . . . . . . . . . . . . .213
Pointing Out the Basics of Singular Points ...............................................213
Finding singular points ......................................................................214
The behavior of singular points .......................................................214
Regular versus irregular singular points.........................................215
Exploring Exciting Euler Equations ...........................................................219
Real and distinct roots.......................................................................220
Real and equal roots ..........................................................................222
Complex roots.....................................................................................223
Putting it all together with a theorem..............................................224
Figuring Series Solutions Near Regular Singular Points..........................225
Identifying the general solution........................................................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
Table of Contents xv
Chapter 11: Working with Laplace Transforms . . . . . . . . . . . . . . . . . .239
Breaking Down a Typical Laplace Transform...........................................239
Deciding Whether a Laplace Transform Converges ................................240
Calculating Basic Laplace Transforms ......................................................241
The transform of 1..............................................................................242
The transform of eat ............................................................................242
The transform of sin at ......................................................................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
Surveying Step Functions............................................................................261
Defining the step function .................................................................261
Figuring the Laplace transform of the step function .....................262
Chapter 12: Tackling Systems of First Order Linear
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265
Introducing the Basics of Matrices ............................................................266
Setting up a matrix .............................................................................266
Working through the algebra ............................................................267
Examining matrices............................................................................268
Mastering Matrix Operations......................................................................269
Equality................................................................................................269
Addition ...............................................................................................270
Subtraction..........................................................................................270
Multiplication of a matrix and a number.........................................270
Multiplication of two matrices..........................................................270
Multiplication of a matrix and a vector ...........................................271
Identity.................................................................................................272
The inverse of a matrix......................................................................272
Having Fun with Eigenvectors ’n’ Things..................................................278
Linear independence .........................................................................278
Eigenvalues and eigenvectors ..........................................................281
Solving Systems of First-Order Linear Homogeneous
Differential Equations ..............................................................................283
Understanding the basics..................................................................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
Crunching the numbers.....................................................................290
Winding up your work .......................................................................292
xvi Differential Equations For Dummies
Chapter 13: 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
Coming up with new code.................................................................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
Chapter 14: 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
Kardi Teknomo’s Page .................................................................................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
Paul’s Online Math Notes ............................................................................319
S.O.S. Math ....................................................................................................319
University of Surrey Tutorial ......................................................................320
Chapter 15: Ten Really Cool Online Differential
Equation Solving Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .321
AnalyzeMath.com’s Runge-Kutta Method Applet ....................................321
Coolmath.com’s Graphing Calculator .......................................................321
Direction Field Plotter .................................................................................322
An Equation Solver from QuickMath Automatic Math Solutions...........322
First Order Differential Equation Solver....................................................322
GCalc Online Graphing Calculator .............................................................322
JavaView Ode Solver....................................................................................323
Math @ CowPi’s System Solver...................................................................323
A Matrix Inverter from QuickMath Automatic Math Solutions ..............323
Visual Differential Equation Solving Applet..............................................323
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