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