CHAPTER 1 INTRODUCTION 

1  (36) 


1  (5) 

1.2 Solutions and Initial Value Problems 


6  (10) 


16  (8) 

1.4 The Approximation Method of Euter 


24  (6) 


30  (1) 

Technical Writing Exercises 


30  (1) 

Group Projects for Chapter 1 


31  (6) 


31  (1) 


32  (1) 


33  (1) 


34  (3) 
CHAPTER 2 FIRST ORDER DIFFERENTIAL EQUATIONS 

37  (50) 

2.1 Introduction: Motion of a Falling Body 


37  (3) 


40  (9) 


49  (9) 


58  (10) 

2.5 Special Integrating Factors 


68  (4) 

2.6 Substitutions and Transformations 


72  (8) 


80  (1) 


81  (1) 

Technical Writing Exercises 


82  (1) 

Group Projects for Chapter 2 


83  (4) 

A. Torricelli's Law of Fluid Flow 


83  (1) 


84  (1) 


84  (1) 

D. Clairaut Equations and Singular Solutions 


85  (1) 

E. Asymptotic Behavior of Solutions to Linear Equations 


86  (1) 
CHAPTER 3 MATHEMATICAL MODELS AND NUMERICAL METHODS INVOLVING FIRST ORDER EQUATIONS 

87  (67) 

3.1 Mathematical Modeling 


87  (2) 

3.2 Compartmental Analysis 


89  (12) 

3.3 Heating and Cooling of Buildings 


101  (7) 


108  (11) 


119  (4) 

3.6 Improved Euler's Method 


123  (11) 

3.7 HigherOrder Numerical Methods: Taylor and RungeKutta 


134  (10) 

Group Projects for Chapter 3 


144  (10) 


144  (1) 


145  (1) 

C. Aircraft Guidance in a Crosswind 


146  (1) 

D. Feedback and the Op Amp 


147  (1) 


148  (1) 

F. Price, Supply, and Demand 


149  (1) 

G. Stability of Numerical Methods 


150  (1) 

H. Period Doubling and Chaos 


151  (3) 
CHAPTER 4 LINEAR SECONDORDER EQUATIONS 


4.1 Introduction: The MassSpring Oscillator 


154  (6) 

4.2 Homogeneous Linear Equations: The General Solution 


160  (9) 

4.3 Auxiliary Equations with Complex Roots 


169  (10) 

4.4 Nonhomgeneous Equations: The Method of Undetermined Coefficients 


179  (7) 

4.5 The Superposition Principle and Undetermined Coefficients Revisited 


186  (8) 

4.6 Variation of Parameters 


194  (4) 

4.7 Qualitative Considerations for VariableCoefficient and Nonlinear Equations 


198  (12) 

4.8 A Closer Look at Free Mechanical Vibrations 


210  (10) 

4.9 A Closer Look at Forced Mechanical Vibrations 


220  (8) 


228  (2) 


230  (1) 

Technical Writing Exercises 


231  (1) 

Group Projects for Chapter 4 


232  (9) 

A. Undetermined Coefficients Using Complex Arithmetic 


232  (1) 

B. An Alternative to the Method of Undetermined Coefficients 


233  (1) 


234  (1) 

D. Linearization of Nonlinear Problems 


235  (1) 

E. Nonlinear Equations Solvable by FirstOrder Techniques 


236  (1) 


237  (1) 


238  (1) 

H. Asymptotic Behavior of Solutions 


239  (2) 
CHAPTER 5 INTRODUCTION TO SYSTEMS AND PHASE PLANE ANALYSIS 

241  (76) 

5.1 Interconnected Fluid Tanks 


241  (2) 

5.2 Elimination Method for Systems with Constant Coefficients 


243  (10) 

5.3 Solving Systems and HigherOrder Equations Numerically 


253  (11) 

5.4 Introduction to the Phase Plane 


264  (15) 

5.5 Coupled MassSpring Systems 


279  (7) 


286  (6) 

5.7 Dynamical Systems, Poincare Maps, and Chaos 


292  (11) 


303  (1) 


304  (2) 

Group Projects for Chapter 5 


306  (11) 


306  (2) 

B. Designing a Landing System for Interplanetary Travel 


308  (1) 


309  (2) 

D. Periodic Solutions to VolterraLotka Systems 


311  (1) 


312  (2) 

F. Strange Behavior of Competing SpeciesPart I 


314  (1) 

G. Cleaning Up the Great Lakes 


315  (2) 
CHAPTER 6 THEORY OF HIGHERORDER LINEAR DIFFERENTIAL EQUATIONS 

317  (31) 

6.1 Basic Theory of Linear Differential Equations 


317  (9) 

6.2 Homogeneous Linear Equations with Constant Coefficients 


326  (7) 

6.3 Undetermined Coefficients and the Annihilator Method 


333  (5) 

6.4 Method of Variation of Parameters 


338  (5) 


343  (1) 


344  (1) 

Technical Writing Exercises 


345  (1) 

Group Projects for Chapter 6 


346  (2) 

A. Justifying the Method of Undetermined Coefficients 


346  (1) 

B. Transverse Vibrations of a Beam 


346  (2) 
CHAPTER 7 LAPLACE TRANSFORMS 

348  (77) 

7.1 Introduction: A Mixing Problem 


348  (3) 

7.2 Definition of the Laplace Transform 


351  (9) 

7.3 Properties of the Laplace Transform 


360  (6) 

7.4 Inverse Laplace Transform 


366  (10) 

7.5 Solving Initial Value Problems 


376  (8) 

7.6 Transforms of Discontinuous and Periodic Functions 


384  (14) 


398  (9) 

7.8 Impulses and the Dirac Delta Function 


407  (7) 

7.9 Solving Linear Systems with Laplace Transforms 


414  (3) 


417  (1) 


418  (1) 

Technical Writing Exercises 


419  (2) 

Group Projects for Chapter 7 


421  (4) 


421  (1) 

B. Frequency Response Modeling 


422  (2) 

C. Determining System Parameters 


424  (1) 
CHAPTER 8 SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS 

425  (78) 

8.1 Introduction: The Taylor Polynomial Approximation 


425  (6) 

8.2 Power Series and Analytic Functions 


431  (9) 

8.3 Power Series Solutions to Linear Differential Equations 


440  (11) 

8.4 Equations with Analytic Coefficients 


451  (6) 

8.5 CauchyEuler (Equidimensional) Equations 


457  (4) 


461  (12) 

8.7 Finding a Second Linearly Independent Solution 


473  (11) 


484  (12) 


496  (1) 


497  (1) 

Technical Writing Exercises 


498  (1) 

Group Projects for Chapter 8 


499  (4) 

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


499  (1) 


500  (1) 


500  (1) 

D. Aging Spring and Bessel Functions 


501  (2) 
CHAPTER 9 MATRIX METHODS FOR LINEAR SYSTEMS 

503  (73) 


503  (5) 

9.2 Review 1: Linear Algebraic Equations 


508  (4) 

9.3 Review 2: Matrices and Vectors 


512  (12) 

9.4 Linear Systems in Normal Form 


524  (9) 

9.5 Homogeneous Linear Systems with Constant Coefficients 


533  (12) 


545  (6) 

9.7 Nonhomogeneous Linear Systems 


551  (7) 

9.8 The Matrix Exponential Function 


558  (9) 


567  (3) 


570  (1) 

Technical Writing Exercises 


571  (1) 

Group Projects for Chapter 9 


572  (4) 

A. Uncoupling Normal Systems 


572  (1) 

B. Matrix Laplace Transform Method 


572  (2) 

C. Undamped SecondOrder Systems 


574  (1) 

D. Strange Behavior of Competing SpeciesPart II 


575  (1) 
CHAPTER 10 PARTIAL DIFFERENTIAL EQUATIONS 

576  (85) 

10.1 Introduction: A Model for Heat Flow 


576  (3) 

10.2 Method of Separation of Variables 


579  (10) 


589  (18) 

10.4 Fourier Cosine and Sine Series 


607  (5) 


612  (13) 


625  (13) 


638  (13) 


651  (2) 

Technical Writing Exercises 


653  (1) 

Group Projects for Chapter 10 


654  (7) 

A. SteadyState Temperature Distribution in a Circular Cylinder 


654  (1) 

B. A Laplace Transform Solution of the Wave Equation 


655  (1) 


656  (2) 

D. Numerical Method for Δu = f on a Rectangle 


658  (3) 
CHAPTER 11 EIGENVALUE PROBLEMS AND STURMLIOUVILLE EQUATIONS 

661  (76) 

11.1 Introduction: Heat Flow in a Nonuniform Wire 


661  (2) 

11.2 Eigenvalues and Eigenfunctions 


663  (9) 

11.3 Regular SturmLiouville Boundary Value Problems 


672  (12) 

11.4 Nonhomogeneous Boundary Value Problems and the Fredholm Alternative 


684  (9) 

11.5 Solution by Eigenfunction Expansion 


693  (6) 


699  (9) 

11.7 Singular SturmLiouville Boundary Value Problems 


708  (9) 

11.8 Oscillation and Comparison Theory 


717  (9) 


726  (3) 


729  (1) 

Technical Writing Exercises 


730  (1) 

Group Projects for Chapter 11 


731  (6) 

A. Hermite Polynomials and the Harmonic Oscillator 


731  (1) 

B. Continuous and Mixed Spectra 


731  (1) 

C. Picone Comparison Theorem 


732  (1) 


733  (1) 

E. FiniteDifference Method for Boundary Value Problems 


734  (3) 
CHAPTER 12 STABILITY OF AUTONOMOUS SYSTEMS 

737  (69) 

12.1 Introduction: Competing Species 


737  (4) 

12.2 Linear Systems in the Plane 


741  (13) 

12.3 Almost Linear Systems 


754  (12) 


766  (9) 

12.5 Lyapunov's Direct Method 


775  (9) 

12.6 Limit Cycles and Periodic Solutions 


784  (9) 

12.7 Stability of HigherDimensional Systems 


793  (6) 


799  (2) 


801  (1) 

Technical Writing Exercises 


802  (1) 

Group Projects for Chapter 12 


803  (3) 

A. Solitons and Kortewegde Vries Equation 


803  (1) 


803  (1) 

C. Computing Phase Plane Diagrams 


804  (1) 

D. Ecosystem on Planet GLIA2 


805  (1) 
CHAPTER 13 EXISTENCE AND UNIQUENESS THEORY 

806  

13.1 Introduction: Successive Approximations 


806  (7) 

13.2 Picard's Existence and Uniqueness Theorem 


813  (8) 

13.3 Existence of Solutions of Linear Equations 


821  (6) 

13.4 Continuous Dependence of Solutions 


827  (7) 


834  (1) 


835  (1) 

Technical Writing Exercises 


835  
APPENDICES 

A1  


A1  


A3  


A5  

D. Method of Least Squares 


A6  

E. RungeKutta Procedure for n Equations 


A9  
ANSWERS TO ODDNUMBERED PROBLEMS 

B1  
INDEX 

I1  