Preface 

vii  


1  (28) 

Some Basic Mathematical Models; Direction Fields 


1  (8) 

Solutions of Some Differential Equations 


9  (8) 

Classification of Differential Equations 


17  (6) 


23  (6) 

First Order Differential Equations 


29  (100) 

Linear Equations with Variable Coefficients 


29  (11) 


40  (7) 

Modeling with First Order Equations 


47  (17) 

Differences Between Linear and Nonlinear Equations 


64  (10) 

Autonomous Equations and Population Dynamics 


74  (15) 

Exact Equations and Integrating Factors 


89  (7) 

Numerical Approximations: Euler's Method 


96  (9) 

The Existence and Uniqueness Theorem 


105  (10) 

First Order Difference Equations 


115  (14) 

Second Order Linear Equations 


129  (80) 

Homogeneous Equations with Constant Coefficients 


129  (8) 

Fundamental Solutions of Linear Homogeneous Equations 


137  (10) 

Linear Independence and the Wronskian 


147  (6) 

Complex Roots of the Characteristic Equation 


153  (7) 

Repeated Roots; Reduction of Order 


160  (9) 

Nonhomogeneous Equations; Method of Undetermined Coefficients 


169  (10) 


179  (7) 

Mechanical and Electrical Vibrations 


186  (14) 


200  (9) 

Higher Order Linear Equations 


209  (22) 

General Theory of nth Order Linear Equations 


209  (5) 

Homogeneous Equations with Constant Coeffients 


214  (8) 

The Method of Undetermined Coefficients 


222  (4) 

The Method of Variation of Parameters 


226  (5) 

Series Solutions of Second Order Linear Equations 


231  (62) 


231  (7) 

Series Solutions near an Ordinary Point, Part I 


238  (11) 

Series Solutions near an Ordinary Point, Part II 


249  (6) 


255  (5) 


260  (7) 

Series Solutions near a Regular Singular Point, Part I 


267  (5) 

Series Solutions near a Regular Singular Point, Part II 


272  (8) 


280  (13) 


293  (46) 

Definition of the Laplace Transform 


293  (6) 

Solution of Initial Value Problems 


299  (11) 


310  (7) 

Differential Equations with Discontinuous Forcing Functions 


317  (7) 


324  (6) 


330  (9) 

Systems of First Order Linear Equations 


339  (80) 


339  (9) 


348  (9) 

Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 


357  (11) 

Basic Theory of Systems of First Order Linear Equations 


368  (5) 

Homogeneous Linear Systems with Constant Coefficients 


373  (11) 


384  (9) 


393  (8) 


401  (10) 

Nonhomogeneous Linear Systems 


411  (8) 


419  (40) 

The Euler or Tangent Line Method 


419  (11) 

Improvements on the Euler Method 


430  (5) 


435  (4) 


439  (6) 

More on Errors; Stability 


445  (10) 

Systems of First Order Equations 


455  (4) 

Nonlinear Differential Equations and Stability 


459  (82) 

The Phase Plane; Linear Systems 


459  (12) 

Autonomous Systems and Stability 


471  (8) 


479  (12) 


491  (12) 


503  (8) 


511  (10) 

Periodic Solutions and Limit Cycles 


521  (11) 

Chaos and Strange Attractors; the Lorenz Equations 


532  (9) 
Answer to Problems 

541  (46) 
Index 

587  