Preface | p. vii |
Introduction | p. 1 |
Some Basic Mathematical Models; Direction Fields | p. 1 |
Solutions of Some Differential Equations | p. 9 |
Classification of Differential Equations | p. 17 |
Historical Remarks | p. 23 |
First Order Differential Equations | p. 29 |
Linear Equations with Variable Coefficients | p. 29 |
Separable Equations | p. 40 |
Modeling with First Order Equations | p. 47 |
Differences Between Linear and Nonlinear Equations | p. 64 |
Autonomous Equations and Population Dynamics | p. 74 |
Exact Equations and Integrating Factors | p. 89 |
Numerical Approximations: Euler's Method | p. 96 |
The Existence and Uniqueness Theorem | p. 105 |
First Order Difference Equations | p. 115 |
Second Order Linear Equations | p. 129 |
Homogeneous Equations with Constant Coefficients | p. 129 |
Fundamental Solutions of Linear Homogeneous Equations | p. 137 |
Linear Independence and the Wronskian | p. 147 |
Complex Roots of the Characteristic Equation | p. 153 |
Repeated Roots; Reduction of Order | p. 160 |
Nonhomogeneous Equations; Method of Undetermined Coefficients | p. 169 |
Variation of Parameters | p. 179 |
Mechanical and Electrical Vibrations | p. 186 |
Forced Vibrations | p. 200 |
Higher Order Linear Equations | p. 209 |
General Theory of nth Order Linear Equations | p. 209 |
Homogeneous Equations with Constant Coeffients | p. 214 |
The Method of Undetermined Coefficients | p. 222 |
The Method of Variation of Parameters | p. 226 |
Series Solutions of Second Order Linear Equations | p. 231 |
Review of Power Series | p. 231 |
Series Solutions near an Ordinary Point, Part I | p. 238 |
Series Solutions near an Ordinary Point, Part II | p. 249 |
Regular Singular Points | p. 255 |
Euler Equations | p. 260 |
Series Solutions near a Regular Singular Point, Part I | p. 267 |
Series Solutions near a Regular Singular Point, Part II | p. 272 |
Bessel's Equation | p. 280 |
The Laplace Transform | p. 293 |
Definition of the Laplace Transform | p. 293 |
Solution of Initial Value Problems | p. 299 |
Step Functions | p. 310 |
Differential Equations with Discontinuous Forcing Functions | p. 317 |
Impulse Functions | p. 324 |
The Convolution Integral | p. 330 |
Systems of First Order Linear Equations | p. 339 |
Introduction | p. 339 |
Review of Matrices | p. 348 |
Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors | p. 357 |
Basic Theory of Systems of First Order Linear Equations | p. 368 |
Homogeneous Linear Systems with Constant Coefficients | p. 373 |
Complex Eigenvalues | p. 384 |
Fundamental Matrices | p. 393 |
Repeated Eigenvalues | p. 401 |
Nonhomogeneous Linear Systems | p. 411 |
Numerical Methods | p. 419 |
The Euler or Tangent Line Method | p. 419 |
Improvements on the Euler Method | p. 430 |
The Runge-Kutta Method | p. 435 |
Multistep Methods | p. 439 |
More on Errors; Stability | p. 445 |
Systems of First Order Equations | p. 455 |
Nonlinear Differential Equations and Stability | p. 459 |
The Phase Plane; Linear Systems | p. 459 |
Autonomous Systems and Stability | p. 471 |
Almost Linear Systems | p. 479 |
Competing Species | p. 491 |
Predator-Prey Equations | p. 503 |
Liapunov's Second Method | p. 511 |
Periodic Solutions and Limit Cycles | p. 521 |
Chaos and Strange Attractors; the Lorenz Equations | p. 532 |
Partial Differential Equations and Fourier Series | p. 541 |
Two-Point Boundary Valve Problems | p. 541 |
Fourier Series | p. 547 |
The Fourier Convergence Theorem | p. 558 |
Even and Odd Functions | p. 564 |
Separation of Variables; Heat Conduction in a Rod | p. 573 |
Other Heat Conduction Problems | p. 581 |
The Wave Equation; Vibrations of an Elastic String | p. 591 |
Laplace's Equation | p. 604 |
Derivation of the Heat Conduction Equation | p. 614 |
Derivation of the Wave Equation | p. 617 |
Boundary Value Problems and Sturm-Liouville Theory | p. 621 |
The Occurrence of Two Point Boundary Value Problems | p. 621 |
Sturm-Liouville Boundary Value Problems | p. 629 |
Nonhomogeneous Boundary Value Problems | p. 641 |
Singular Sturm-Liouville Problems | p. 656 |
Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion | p. 663 |
Series of Orthogonal Functions: Mean Convergence | p. 669 |
Answers to Problems | p. 679 |
Index | p. 737 |
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