

1  (41) 


1  (5) 

Solutions and Initial Value Problems 


6  (10) 


16  (8) 


24  (7) 

The Approximation Method of Euler 


31  (11) 


37  (1) 

Technical Writing Exercises 


37  (1) 

Group Projects for Chapter 1 


38  (1) 


38  (1) 


39  (1) 


40  (2) 

First Order Differential Equations 


42  (51) 

Introduction: Motion of a Falling Body 


42  (3) 


45  (9) 


54  (9) 


63  (10) 

Special Integrating Factors 


73  (4) 

Substitutions and Transformations 


77  (16) 


86  (1) 


87  (1) 

Technical Writing Exercises 


88  (1) 

Group Projects for Chapter 2 


89  (1) 


89  (1) 


89  (1) 

Asymptotic Behavior of Solutions to Linear Equations 


90  (1) 

Torricelli's Law of Fluid Flow 


90  (2) 

Clairaut Equations and Singular Solutions 


92  (1) 

Mathematical Models and Numerical Methods Involving First Order Equations 


93  (63) 


93  (2) 


95  (12) 

Heating and Cooling of Buildings 


107  (7) 


114  (10) 


124  (11) 

HigherOrder Numerical Methods: Taylor and RungeKutta 


135  (9) 

Professional Codes for Solving Initial Value Problems 


144  (12) 

Group Projects for Chapter 3 


148  (1) 

Delay Differential Equations 


148  (1) 


149  (1) 


150  (1) 

Aircraft Guidance in a Crosswind 


151  (1) 

Stability of Numerical Methods 


152  (1) 

Period Doubling and Chaos 


153  (2) 


155  (1) 

Linear Second Order Equations 


156  (105) 

Introduction: The MassSpring Oscillator 


156  (5) 

Linear Differential Operators 


161  (7) 

Fundamental Solutions of Homogeneous Equations 


168  (10) 


178  (5) 

Homogeneous Linear Equations with Constant Coefficients 


183  (8) 

Auxiliary Equations with Complex Roots 


191  (9) 

Superposition and Nonhomogeneous Equations 


200  (4) 

Method of Undetermined Coefficients 


204  (9) 


213  (5) 

Qualitative Considerations for VariableCoefficient and Nonlinear Equations 


218  (12) 

A Closer Look at Free Mechanical Vibrations 


230  (10) 

A Closer Look at Forced Mechanical Vibrations 


240  (21) 


248  (2) 


250  (1) 

Technical Writing Exercises 


251  (1) 

Group Projects for Chapter 4 


252  (1) 

Undetermined Coefficients Using Complex Arithmetic 


252  (1) 

An Alternative to the Method of Undetermined Coefficients 


253  (1) 


254  (1) 

Linearization of Nonlinear Problems 


255  (1) 

Nonlinear Equations Solvable by First Order Techniques 


256  (1) 


257  (1) 


258  (1) 

Asymptotic Behavior of Solutions 


259  (2) 

Introduction to Systems and Phase Plane Analysis 


261  (77) 

Interconnected Fluid Tanks 


261  (2) 

Introduction to the Phase Plane 


263  (15) 

Elimination Method for Systems 


278  (8) 

Coupled MassSpring Systems 


286  (7) 


293  (8) 

Numerical Methods for HigherOrder Equations and Systems 


301  (14) 

Dynamical Systems, Poincare Maps, and Chaos 


315  (23) 


325  (2) 


327  (1) 

Group Projects for Chapter 5 


328  (1) 

Designing a Landing System for Interplanetary Travel 


328  (1) 


329  (1) 

Effects of Hunting on PredatorPrey Systems 


330  (1) 

Periodic Solutions to VolterraLotka Systems 


331  (1) 


332  (2) 


334  (1) 

Strange Behavior of Competing SpeciesPart I 


335  (1) 

Cleaning Up the Great Lakes 


336  (2) 

Theory of HigherOrder Linear Differential Equations 


338  (31) 

Basic Theory of Linear Differential Equations 


338  (9) 

Homogeneous Linear Equations with Constant Coefficients 


347  (7) 

Undetermined Coefficients and the Annihilator Method 


354  (6) 

Method of Variation of Parameters 


360  (9) 


364  (2) 


366  (1) 

Technical Writing Exercises 


366  (1) 

Group Projects for Chapter 6 


367  (1) 

Justifying the Method of Undetermined Coefficients 


367  (1) 

Transverse Vibrations of Beam 


367  (2) 


369  (78) 

Introduction: A Mixing Problem 


369  (4) 

Definition of the Laplace Transform 


373  (9) 

Properties of the Laplace Transform 


382  (6) 

Inverse Laplace Transform 


388  (10) 

Solving Initial Value Problems 


398  (8) 

Transforms of Discontinuous and Periodic Functions 


406  (14) 


420  (9) 

Impulses and the Dirac Delta Function 


429  (7) 

Solving Linear Systems with Laplace Transforms 


436  (11) 


439  (1) 


440  (1) 

Technical Writing Exercises 


441  (2) 

Group Projects for Chapter 7 


443  (1) 


443  (1) 

Frequency Response Modeling 


444  (2) 

Determining System Parameters 


446  (1) 

Series Solutions of Differential Equations 


447  (80) 

Introduction: The Taylor Polynomial Approximation 


447  (6) 

Power Series and Analytic Functions 


453  (9) 

Power Series Solutions to Linear Differential Equations 


462  (11) 

Equations with Analytic Coefficients 


473  (6) 

CauchyEuler (Equidimensional) Equations Revisited 


479  (4) 


483  (12) 

Finding a Second Linearly Independent Solution 


495  (12) 


507  (20) 


520  (1) 


521  (1) 

Technical Writing Exercises 


522  (1) 

Group Projects for Chapter 8 


523  (1) 

Spherically Symmetric Solutions to Schrodinger's Equation for the Hydrogen Atom 


523  (1) 


524  (1) 


524  (1) 

Aging Spring and Bessel Functions 


525  (2) 

Matrix Methods for Linear Systems 


527  (73) 


527  (5) 

Linear Algebraic Equations 


532  (4) 


536  (12) 

Linear Systems in Normal Form 


548  (9) 

Homogeneous Linear Systems with Constant Coefficients 


557  (12) 


569  (6) 

Nonhomogeneous Linear Systems 


575  (7) 

The Matrix Exponential Function 


582  (18) 


591  (3) 


594  (1) 

Technical Writing Exercises 


595  (1) 

Group Projects for Chapter 9 


596  (1) 

Uncoupling Normal Systems 


596  (1) 

Matrix Laplace Transform Method 


596  (2) 

Undamped Second Order Systems 


598  (1) 

Strange Behavior of Competing SpeciesPart II 


599  (1) 

Partial Differential Equations 


600  

Introduction: A Model for Heat Flow 


600  (3) 

Method of Separation of Variables 


603  (10) 


613  (18) 

Fourier Cosine and Sine Series 


631  (5) 


636  (13) 


649  (13) 


662  


675  (2) 

Technical Writing Exercises 


677  (1) 

Group Projects for Chapter 10 


678  (1) 

SteadyState Temperature Distribution in a Circular Cylinder 


678  (1) 

A Laplace Transform Solution of the Wave Equation 


679  (1) 


680  (2) 

Numerical Method for Δu = f on a Rectangle 


682  
APPENDICES 

A1  (1) 


A1  (1) 


A3  (1) 


A5  (1) 

D. Method of Least Squares 


A6  (1) 

E. RungeKutta Procedure for n Equations 


A9  (1) 
Answers to OddNumbered Problems 

B1  (1) 
Index 

I1  