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| (Most chapters end with a Chapter Summary, Review Problems and Group Projects.) | |
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| Introduction | |
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| Background | |
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| Solutions and Initial Value Problems | |
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| Direction Fields | |
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| The Approximation Method of Euler | |
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| First Order Differential Equations | |
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| Introduction: Motion of a Falling Body | |
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| Separable Equations | |
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| Linear Equations | |
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| Exact Equations | |
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| Special Integrating Factors | |
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| Substitutions and Transformations | |
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| Mathematical Models and Numerical Methods Involving First Order Equations | |
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| Mathematical Modeling | |
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| Compartmental Analysis | |
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| Heating and Cooling of Buildings | |
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| Newtonian Mechanics | |
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| Electrical Circuits | |
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| Improved Euler's Method | |
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| Higher-Order Numerical Methods: Taylor and Runge-Kutta | |
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| Linear Second Order Equations | |
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| Introduction: The Mass-Spring Oscillator | |
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| Homogeneous Linear Equations | |
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| The General Solution | |
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| Auxiliary Equations with Complex Roots | |
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| Nonhomogeneous Equations: the Method of Undetermined Coefficients | |
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| The Superposition Principle and Undetermined Coefficients Revisited | |
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| Variation of Parameters | |
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| Qualitative Considerations for Variable-Coefficient and Nonlinear Equations | |
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| A Closer Look at Free Mechanical Vibrations | |
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| A Closer Look at Forced Mechanical Vibrations | |
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| Introduction to Systems and Phase Plane Analysis | |
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| Interconnected Fluid Tanks | |
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| Elimination Method for Systems with Constant Coefficients | |
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| Solving Systems and Higher-Order Equations Numerically | |
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| Introduction to the Phase Plane | |
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| Coupled Mass-Spring Systems | |
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| Electrical Systems | |
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| Dynamical Systems, Poincaré Maps, and Chaos | |
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| Theory of Higher-Order Linear Differential Equations | |
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| Basic Theory of Linear Differential Equations | |
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| Homogeneous Linear Equations with Constant Coefficients | |
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| Undetermined Coefficients and the Annihilator Method | |
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| Method of Variation of Parameters | |
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| Laplace Transforms | |
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| Introduction: A Mixing Problem | |
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| Definition of the Laplace Transform | |
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| Properties of the Laplace Transform | |
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| Inverse Laplace Transform | |
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| Solving Initial Value Problems | |
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| Transforms of Discontinuous and Periodic Functions | |
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| Convolution | |
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| Impulses and the Dirac Delta Function | |
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| Solving Linear Systems with Laplace Transforms | |
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| Series Solutions of Differential Equations | |
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| Introduction: The Taylor Polynomial Approximation | |
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| Power Series and Analytic Functions | |
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| Power Series Solutions to Linear Differential Equations | |
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| Equations with Analytic Coefficients | |
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| Cauchy-Euler (Equidimensional) Equations | |
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| Method of Frobenius | |
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| Finding a Second Linearly Independent Solution | |
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| Special Functions | |
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| Matrix Methods for Linear Systems | |
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| Introduction | |
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| Linear Algebraic Equations | |
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| Matrices and Vectors | |
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| Linear Systems in Normal Form | |
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| Homogeneous Linear Systems with Constant Coefficients | |
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| Complex Eigenvalues | |
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| Nonhomogeneous Linear Systems | |
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| The Matrix Exponential Function | |
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| Partial Differential Equations | |
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| Introduction: A Model for Heat Flow | |
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| Method of Separation of Variables | |
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| Fourier Series | |
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| Fourier Cosine and Sine Series | |
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| The Heat Equation | |
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| The Wave Equation | |
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| Laplace's Equation | |
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| Appendices | |
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| Newton's Method | |
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| Simpson's Rule | |
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| Cramer's Rule | |
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| Method of Least Squares | |
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| Runge-Kutta Precedure for n | |
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| Equations | |
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| Answers to Odd-Numbered Problems | |
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| Index | |