Preface 

ix  

Introduction to Differential Equations 


1  (17) 

Differential Equation Models 


2  (5) 


7  (3) 


10  (8) 


18  (105) 

Differential Equations and Solutions 


18  (13) 

Solutions to Separable Equations 


31  (13) 


44  (10) 


54  (10) 


64  (9) 

Exact Differential Equations 


73  (17) 

Existence and Uniqueness of Solutions 


90  (12) 

Dependence of Solutions on Initial Conditions 


102  (5) 

Autonomous Equations and Stability 


107  (16) 

Project 2.10 The Daredevil Skydiver 


120  (3) 

Modeling and Applications 


123  (40) 

Modeling Population Growth 


124  (14) 

Models and the Real World 


138  (5) 


143  (9) 


152  (11) 

Project 3.5 The Spruce Budworm 


158  (3) 

Project 3.6 Social Security, Now or Later 


161  (2) 


163  (68) 


163  (11) 

SecondOrder Equations and Systems 


174  (5) 

Linear, Homogeneous Equations with Constant Coefficients 


179  (11) 


190  (9) 

Inhomogeneous Equations; the Method of Undetermined Coefficients 


199  (10) 


209  (6) 


215  (16) 

Project 4.8 Nonlinear Oscillators 


228  (3) 


231  (70) 

The Definition of the Laplace Transform 


232  (9) 

Basic Properties of the Laplace Transform 


241  (7) 

The Inverse Laplace Transform 


248  (8) 

Using the Laplace Transform to Solve Differential Equations 


256  (10) 

Discontinuous Forcing Terms 


266  (14) 


280  (7) 


287  (11) 


298  (3) 

Project 5.9 Forced Harmonic Oscillators 


299  (2) 


301  (34) 


302  (12) 


314  (8) 

Numerical Error Comparisons 


322  (5) 


327  (5) 


332  (3) 

Project 6.6 Numerical Error Comparison 


334  (1) 


335  (61) 


335  (13) 

The Geometry of Systems of Linear Equations 


348  (5) 

Solving Systems of Equations 


353  (9) 

Properties of Solution Sets 


362  (9) 


371  (13) 


384  (12) 

An Introduction to Systems 


396  (48) 


396  (9) 

Geometric Interpretation of Solutions 


405  (12) 


417  (8) 


425  (19) 

Project 8.5 LongTerm Behavior of Solutions 


441  (3) 

Linear Systems with Constant Coefficents 


444  (101) 

Overview of the Technique 


444  (8) 


452  (14) 


466  (18) 

Higher Dimensional Systems 


484  (8) 

The Exponential of a Matrix 


492  (18) 

Qualitative Analysis of Lienar Systems 


510  (5) 

HigherOrder Linear Equations 


515  (13) 

Inhomogeneous Linear Systems 


528  (17) 

Project 9.9 Phase Plane Portraits 


538  (1) 

Project 9.10 Oscillations of Linear Molecules 


539  (6) 


545  (92) 

The Linearization of a Nonlinear System 


545  (14) 

LongTerm Behavior of Solutions 


559  (7) 

Invariant Sets and the Use of Nullclines 


566  (8) 

LongTerm Behavior of Solutions to Planar Systems 


574  (12) 


586  (6) 


592  (17) 


609  (10) 


619  (18) 

Project 10.9 Human Immune Response to Infectious Disease 


631  (3) 

Project 10.10 Analysis of Competing Species 


634  (3) 

Series Solutions to Differential Equations 


637  (75) 


638  (12) 

Series Solutions Near Ordinary Points 


650  (12) 


662  (6) 

Types of Singular PointsEuler's Equation 


668  (9) 

Series Solutions Near Regular Singular Points 


677  (13) 

Solutions in the Exceptional Cases 


690  (11) 

Bessel's Equation and Bessel Functions 


701  (11) 


712  (38) 

Computation of Fourier Series 


713  (11) 

Convergence of Fourier Series 


724  (9) 

Fourier Cosine and Sine Series 


733  (5) 

The Complex Form of a Fourier Series 


738  (3) 

The Discrete Fourier Transform and the FFT 


741  (9) 

Partial Differential Equations 


750  

Derivation of the Heat Equation 


750  

Separation of Variables for the Heat Equation 


756  


768  


778  

Laplace's Equation on a Disk 


785  


791  

Orthogonality and Generalized Fourier Series 


801  

Temperature in a BallLegendre Polynomials 


810  

Time Dependent PDEs in Higher Dimension 


814  

Domains with Circular SymmetryBessel Functions 


821  
Answers to OddNumbered Problems 

A1  
Index 

I1  