Preface 

xi  

Introduction to Differential Equations 


1  (14) 


1  (1) 

Examples of Differential Equations 


1  (7) 


8  (7) 

First Order Differential Equations 


15  (92) 


15  (4) 

First Order Linear Differential Equations 


19  (12) 

Introduction to Mathematical Models 


31  (10) 

Population Dynamics and Radioactive Decay 


41  (7) 

First Order Nonlinear Differential Equations 


48  (6) 

Separable First Order Equations 


54  (9) 

Exact Differential Equations 


63  (7) 

The Logistic Population Model 


70  (7) 

Applications to Mechanics 


77  (12) 


89  (18) 


100  (1) 


101  (6) 

Second and Higher Order Linear Differential Equations 


107  (106) 


108  (7) 

The General Solution of Homogeneous Equations 


115  (6) 

Constant Coefficient Homogeneous Equations 


121  (6) 

Real Repeated Roots; Reduction of Order 


127  (5) 


132  (10) 

Unforced Mechanical Vibrations 


142  (12) 

The General Solution of a Linear Nonhomogeneous Equation 


154  (4) 

The Method of Undetermined Coefficients 


158  (10) 

The Method of Variation of Parameters 


168  (6) 

Forced Mechanical Vibrations, Electrical Networks, and Resonance 


174  (14) 

Higher Order Linear Homogeneous Differential Equations 


188  (7) 

Higher Order Homogeneous Constant Coefficient Differential Equations 


195  (6) 

Higher Order Linear Nonhomogeneous Differential Equations 


201  (12) 


206  (1) 


206  (7) 

First Order Linear Systems 


213  (104) 


213  (10) 


223  (5) 

Homogeneous Linear Systems 


228  (10) 

Constant Coefficient Homogeneous Systems; the Eigenvalue Problem 


238  (9) 

Real Eigenvalues and the Phase Plane 


247  (9) 


256  (10) 


266  (11) 

Nonhomogeneous Linear Systems 


277  (11) 

Numerical Methods for Systems of Linear Differential Equations 


288  (12) 

The Exponential Matrix and Diagonalization 


300  (17) 


310  (1) 


311  (6) 


317  (74) 


317  (12) 


329  (15) 

The Method of Partial Fractions 


344  (6) 

Laplace Transforms of Periodic Functions and System Transfer Functions 


350  (9) 

Solving Systems of Differential Equations 


359  (9) 


368  (9) 

The Delta Function and Impulse Response 


377  (14) 


385  (6) 


391  (80) 


391  (9) 

Equilibrium Solutions and Direction Fields 


400  (13) 


413  (11) 


424  (9) 

Linearization and the Local Picture 


433  (15) 

TwoDimensional Linear Systems 


448  (10) 

PredatorPrey Population Models 


458  (13) 


466  (5) 


471  (44) 


471  (2) 

Euler's Method, Heun's Method, and the Modified Euler's Method 


473  (6) 


479  (14) 


493  (22) 

Appendix 1: Convergence of OneStep Methods 


506  (1) 

Appendix 2: Stability of OneStep Methods 


507  (3) 


510  (5) 

Series Solutions of Linear Differential Equations 


515  (50) 


515  (12) 

Series Solutions Near an Ordinary Point 


527  (9) 


536  (6) 

Solutions Near a Regular Singular Point and the Method of Frobenius 


542  (8) 

The Method of Frobenius Continued: Special Cases and a Summary 


550  (15) 


561  (4) 

Second Order Partial Differential Equations and Fourier Series 


565  (94) 


565  (5) 

Heat Flow in a Thin Bar; Separation of Variables 


570  (10) 


580  (9) 


589  (11) 


600  (16) 


616  (12) 


628  (13) 

HigherDimensional Problems; Nonhomogeneous Equations 


641  (18) 


655  (4) 

First Order Partial Differential Equations and the Method of Characteristics 


659  (22) 


659  (3) 


662  (6) 


668  (3) 

The Method of Characteristics 


671  (10) 


679  (2) 

Linear TwoPoint Boundary Value Problems 


681  


681  (1) 


682  (11) 

TwoPoint Boundary Value Problems for Linear Systems 


693  (12) 

SturmLiouville Boundary Value Problems 


705  


715  
Answers 

1  (1) 
Index 

1  