Preface |
Introduction / Chapter 1: |
Some Basic Mathematical Models; Direction Fields / 1.1: |
Solutions of Some Differential Equations / 1.2: |
Classification of Differential Equations / 1.3: |
Historical Remarks / 1.4: |
First Order Differential Equations / Chapter 2: |
Linear Equations with Variable Coefficients / 2.1: |
Separable Equations / 2.2: |
Modeling with First Order Equations / 2.3: |
Differences Between Linear and Nonlinear Equations / 2.4: |
Autonomous Equations and Population Dynamics / 2.5: |
Exact Equations and Integrating Factors / 2.6: |
Numerical Approximations: Euler's Method / 2.7: |
The Existence and Uniqueness Theorem / 2.8: |
First Order Difference Equations / 2.9: |
Second Order Linear Equations / Chapter 3: |
Homogeneous Equations with Constant Coefficients / 3.1: |
Fundamental Solutions of Linear Homogeneous Equations / 3.2: |
Linear Independence and the Wronskian / 3.3: |
Complex Roots of the Characteristic Equation / 3.4: |
Repeated Roots; Reduction of Order / 3.5: |
Nonhomogeneous Equations; Method of Undetermined Coefficients / 3.6: |
Variation of Parameters / 3.7: |
Mechanical and Electrical Vibrations / 3.8: |
Forced Vibrations / 3.9: |
Higher Order Linear Equations / Chapter 4: |
General Theory of nth Order Linear Equations / 4.1: |
Homogeneous Equations with Constant Coeffients / 4.2: |
The Method of Undetermined Coefficients / 4.3: |
The Method of Variation of Parameters / 4.4: |
Series Solutions of Second Order Linear Equations / Chapter 5: |
Review of Power Series / 5.1: |
Series Solutions near an Ordinary Point, Part I / 5.2: |
Series Solutions near an Ordinary Point, Part II / 5.3: |
Regular Singular Points / 5.4: |
Euler Equations / 5.5: |
Series Solutions near a Regular Singular Point, Part I / 5.6: |
Series Solutions near a Regular Singular Point, Part II / 5.7: |
Bessel's Equation / 5.8: |
The Laplace Transform / Chapter 6: |
Definition of the Laplace Transform / 6.1: |
Solution of Initial Value Problems / 6.2: |
Step Functions / 6.3: |
Differential Equations with Discontinuous Forcing Functions / 6.4: |
Impulse Functions / 6.5: |
The Convolution Integral / 6.6: |
Systems of First Order Linear Equations / Chapter 7: |
Review of Matrices / 7.1: |
Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors / 7.3: |
Basic Theory of Systems of First Order Linear Equations / 7.4: |
Homogeneous Linear Systems with Constant Coefficients / 7.5: |
Complex Eigenvalues / 7.6: |
Fundamental Matrices / 7.7: |
Repeated Eigenvalues / 7.8: |
Nonhomogeneous Linear Systems / 7.9: |
Numerical Methods / Chapter 8: |
The Euler or Tangent Line Method / 8.1: |
Improvements on the Euler Method / 8.2: |
The Runge-Kutta Method / 8.3: |
Multistep Methods / 8.4: |
More on Errors; Stability / 8.5: |
Systems of First Order Equations / 8.6: |
Nonlinear Differential Equations and Stability / Chapter 9: |
The Phase Plane; Linear Systems / 9.1: |
Autonomous Systems and Stability / 9.2: |
Almost Linear Systems / 9.3: |
Competing Species / 9.4: |
Predator-Prey Equations / 9.5: |
Liapunov's Second Method / 9.6: |
Periodic Solutions and Limit Cycles / 9.7: |
Chaos and Strange Attractors; the Lorenz Equations / 9.8: |
Partial Differential Equations and Fourier Series / Chapter 10: |
Two-Point Boundary Valve Problems / 10.1: |
Fourier Series / 10.2: |
The Fourier Convergence Theorem / 10.3: |
Even and Odd Functions / 10.4: |
Separation of Variables; Heat Conduction in a Rod / 10.5: |
Other Heat Conduction Problems / 10.6: |
The Wave Equation; Vibrations of an Elastic String / 10.7: |
Laplace's Equation / 10.8: |
Derivation of the Heat Conduction Equation / Appendix A.: |
Derivation of the Wave Equation / Appendix B.: |
Boundary Value Problems and Sturm-Liouville Theory / Chapter 11: |
The Occurrence of Two Point Boundary Value Problems / 11.1: |
Sturm-Liouville Boundary Value Problems / 11.2: |
Nonhomogeneous Boundary Value Problems / 11.3: |
Singular Sturm-Liouville Problems / 11.4: |
Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion / 11.5: |
Series of Orthogonal Functions: Mean Convergence / 11.6: |
Answers to Problems |
Index |
Preface |
Introduction / Chapter 1: |
Some Basic Mathematical Models; Direction Fields / 1.1: |