Preface 

vii  
Chapter 1 FirstOrder Differential Equations 

1  (95) 

1.1 Differential Equations and Mathematical Models 


1  (9) 

1.2 Integrals as General and Particular Solutions 


10  (8) 

1.3 Slope Fields and Solution Curves 


18  (13) 

1.4 Separable Equations and Applications 


31  (14) 

1.5 Linear FirstOrder Equations 


45  (12) 

1.6 Substitution Methods and Exact Equations 


57  (14) 


71  (11) 

1.8 AccelerationVelocity Models 


82  (14) 
Chapter 2 Linear Equations of Higher Order 

96  (92) 

2.1 Introduction: SecondOrder Linear Equations 


96  (13) 

2.2 General Solutions of Linear Equations 


109  (11) 

2.3 Homogeneous Equations with Constant Coefficients 


120  (11) 

2.4 Mechanical Vibrations 


131  (13) 

2.5 Nonhomogeneous Equations and Undetermined Coefficients 


144  (13) 

2.6 Forced Oscillations and Resonance 


157  (11) 


168  (8) 

2.8 Endpoint Problems and Eígenvalues 


176  (12) 
Chapter 3 Power Series Methods 

188  (71) 

3.1 Introduction and Review of Power Series 


188  (13) 

3.2 Series Solutions Near Ordinary Points 


201  (11) 

3.3 Regular Singular Points 


212  (15) 

3.4 Method of Frobenius: The Exceptional Cases 


227  (14) 


241  (9) 

3.6 Applications of Bessel Functions 


250  (9) 
Chapter 4 Laplace Transform Methods 

259  (58) 

4.1 Laplace Transforms and Inverse Transforms 


259  (10) 

4.2 Transformation of Initial Value Problems 


269  (11) 

4.3 Translation and Partial Fractions 


280  (7) 

4.4 Derivatives, Integrals, and Products of Transforms 


287  (8) 

4.5 Periodic and Piecewise Continuous Input Functions 


295  (11) 

4.6 Impulses and Delta Functions 


306  (11) 
Chapter 5 Linear Systems of Differential Equations 

317  (103) 

5.1 FirstOrder Systems and Applications 


317  (12) 

5.2 The Method of Elimination 


329  (9) 

5.3 Matrices and Linear Systems 


338  (19) 

5.4 The Eigenvalue Method for Homogeneous Systems 


357  (15) 

5.5 SecondOrder Systems and Mechanical Applications 


372  (11) 

5.6 Multiple Eigenvalue Solutions 


383  (16) 

5.7 Matrix Exponentials and Linear Systems 


399  (12) 

5.8 Nonhomogeneous Linear Systems 


411  (9) 
Chapter 6 Numerical Methods 

420  (50) 

6.1 Numerical Approximation: Euler's Method 


420  (12) 

6.2 A Closer Look at the Euler Method 


432  (11) 

6.3 The RungeKutta Method 


443  (10) 

6.4 Numerical Methods for Systems 


453  (17) 
Chapter 7 Nonlinear Systems and Phenomena 

470  (74) 

7.1 Equilibrium Solutions and Stability 


470  (8) 

7.2 Stability and the Phase Plane 


478  (12) 

7.3 Linear and Almost Linear Systems 


490  (13) 

7.4 Ecological Models: Predators and Competitors 


503  (13) 

7.5 Nonlinear Mechanical Systems 


516  (16) 

7.6 Chaos in Dynamical Systems 


532  (12) 
Chapter 8 Fourier Series Methods 

544  (67) 

8.1 Periodic Functions and Trigonometric Series 


544  (9) 

8.2 General Fourier Series and Convergence 


553  (7) 

8.3 Fourier Sine and Cosine Series 


560  (10) 

8.4 Applications of Fourier Series 


570  (5) 

8.5 Heat Conduction and Separation of Variables 


575  (13) 

8.6 Vibrating Strings and the OneDimensional Wave Equation 


588  (12) 

8.7 SteadyState Temperature and Laplace's Equation 


600  (11) 
Chapter 9 Eigenvalues and Boundary Value Problems 

611  (61) 

9.1 SturmLiouville Problems and Eigenfunction Expansions 


611  (11) 

9.2 Applications of Eigenfunction Series 


622  (9) 

9.3 Steady Periodic Solutions and Natural Frequencies 


631  (8) 

9.4 Cylindrical Coordinate Problems 


639  (14) 

9.5 HigherDimensional Phenomena 


653  (19) 
References for Further Study 

672  (3) 
Appendix: Existence and Uniqueness of Solutions 

675  (14) 
Answers to Selected Problems 

689  
Index 

I1  