Application Modules 

viii  
Preface 

xi  

FirstOrder Differential Equations 


1  (76) 

Differential Equations and Mathematical Models 


1  (9) 

Integrals as General and Particular Solutions 


10  (8) 

Slope Fields and Solution Curves 


18  (13) 

Separable Equations and Applications 


31  (15) 

Linear FirstOrder Equations 


46  (12) 

Substitution Methods and Exact Equations 


58  (19) 

Mathematical Models and Numerical Methods 


77  (67) 


77  (13) 

Equilibrium Solutions and Stability 


90  (8) 

AccelerationVelocity Models 


98  (12) 

Numerical Approximation: Euler's Method 


110  (12) 

A Closer Look at the Euler Method 


122  (10) 


132  (12) 

Linear Equations of Higher Order 


144  (98) 

Introduction: SecondOrder Linear Equations 


144  (14) 

General Solutions of Linear Equations 


158  (12) 

Homogeneous Equations with Constant Coefficients 


170  (12) 


182  (13) 

Nonhomogeneous Equations and Undetermined Coefficients 


195  (14) 

Forced Oscillations and Resonance 


209  (13) 


222  (7) 

Endpoint Problems and Eigenvalues 


229  (13) 

Introduction to Systems of Differential Equations 


242  (39) 

FirstOrder Systems and Applications 


242  (12) 

The Method of Elimination 


254  (11) 

Numerical Methods for Systems 


265  (16) 

Linear Systems of Differential Equations 


281  (85) 

Matrices and Linear Systems 


281  (19) 

The Eigenvalue Method for Homogeneous Systems 


300  (15) 

SecondOrder Systems and Mechanical Applications 


315  (13) 

Multiple Eigenvalue Solutions 


328  (16) 

Matrix Exponentials and Linear Systems 


344  (14) 

Nonhomogeneous Linear Systems 


358  (8) 

Nonlinear Systems and Phenomena 


366  (69) 

Stability and the Phase Plane 


366  (12) 

Linear and Almost Linear Systems 


378  (15) 

Ecological Models: Predators and Competitors 


393  (13) 

Nonlinear Mechanical Systems 


406  (17) 

Chaos in Dynamical Systems 


423  (12) 

Laplace Transform Methods 


435  (62) 

Laplace Transforms and Inverse Transforms 


435  (11) 

Transformation of Initial Value Problems 


446  (11) 

Translation and Partial Fractions 


457  (10) 

Derivatives, Integrals, and Products of Transforms 


467  (8) 

Periodic and Piecewise Continuous Input Functions 


475  (11) 

Impulses and Delta Functions 


486  (11) 


497  (75) 

Introduction and Review of Power Series 


497  (13) 

Series Solutions Near Ordinary Points 


510  (13) 


523  (16) 

Method of Frobenius: The Exceptional Cases 


539  (15) 


554  (9) 

Applications of Bessel Functions 


563  (9) 


572  (73) 

Periodic Functions and Trigonometric Series 


572  (9) 

General Fourier Series and Convergence 


581  (8) 

Fourier Sine and Cosine Series 


589  (12) 

Applications of Fourier Series 


601  (5) 

Heat Conduction and Separation of Variables 


606  (15) 

Vibrating Strings and the OneDimensional Wave Equation 


621  (14) 

SteadyState Temperature and Laplace's Equation 


635  (10) 

Eigenvalues and Boundary Value Problems 


645  (66) 

SturmLiouville Problems and Eigenfunction Expansions 


645  (13) 

Applications of Eigenfunction Series 


658  (10) 

Steady Periodic Solutions and Natural Frequencies 


668  (10) 

Cylindrical Coordinate Problems 


678  (15) 

HigherDimensional Phenomena 


693  (18) 
References for Further Study 

711  (3) 
Appendix: Existence and Uniqueness of Solutions 

714  (15) 
Answers to Selected Problems 

729  
Index 

1  