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Preface | |
Introduction 1 | |
Some Basic Mathematical Models | |
Direction Fields | |
Solutions of Some Differential Equations | |
Classification of Differential Equations | |
Historical Remarks | |
First Order Differential Equations | |
Linear Equations | |
Method of Integrating Factors | |
Separable Equations | |
Modeling with First Order Equations | |
Differences Between Linear and Nonlinear Equations | |
Autonomous Equations and Population Dynamics | |
Exact Equations and Integrating Factors | |
Numerical Approximations: Euler's Method | |
The Existence and Uniqueness Theorem | |
First Order Difference Equations | |
Second Order Linear Equations 135 | |
Homogeneous Equations with Constant Coef?cients | |
Fundamental Solutions of Linear Homogeneous Equations | |
The Wronskian | |
Complex Roots of the Characteristic Equation | |
Repeated Roots | |
Reduction of Order | |
Nonhomogeneous Equations | |
Method of Undetermined Coefficients | |
Variation of Parameters | |
Mechanical and Electrical Vibrations | |
Forced Vibrations | |
Higher Order Linear Equations | |
General Theory of nth Order Linear Equations | |
Homogeneous Equations with Constant Coef?cients | |
The Method of Undetermined Coef?cients | |
The Method of Variation of Parameters | |
Series Solutions of Second Order Linear Equations | |
Review of Power Series | |
Series Solutions Near an Ordinary Point, Part I | |
Series Solutions Near an Ordinary Point, Part II | |
Euler Equations | |
Regular Singular Points | |
Series Solutions Near a Regular Singular Point, Part I | |
Series Solutions Near a Regular Singular Point, Part II | |
Bessel's Equation | |
The Laplace Transform | |
Definition of the Laplace Transform | |
Solution of Initial Value Problems | |
Step Functions | |
Differential Equations with Discontinuous Forcing Functions | |
Impulse Functions | |
The Convolution Integral | |
Systems of First Order Linear Equations | |
Introduction | |
Review of Matrices | |
Systems of Linear Algebraic Equations | |
Linear Independence, Eigenvalues, Eigenvectors | |
Basic Theory of Systems of First Order Linear Equations | |
Homogeneous Linear Systems with Constant Coefficients | |
Complex Eigenvalues | |
Fundamental Matrices | |
Repeated Eigenvalues | |
Nonhomogeneous Linear Systems | |
Numerical Methods | |
The Euler or Tangent Line Method | |
Improvements on the Euler Method | |
The Runge-Kutta Method | |
Multistep Methods | |
More on Errors | |
Stability | |
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