Table of Contents(Most chapters end with a Chapter Summary, Review Problems and Group Projects.)
Solutions and Initial Value Problems.
The Approximation Method of Euler.
2. First Order Differential Equations.
Introduction: Motion of a Falling Body.
Special Integrating Factors.
Substitutions and Transformations.
3. Mathematical Models and Numerical Methods Involving First Order Equations.
Heating and Cooling of Buildings.
Improved Euler's Method.
Higher-Order Numerical Methods: Taylor and Runge-Kutta.
4. 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.
5. Introduction to Systems andPhase 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.
Dynamical Systems, Poincaré Maps, and Chaos.
6. 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.
7. 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.
Impulses and the Dirac Delta Function.
Solving Linear Systems with Laplace Transforms.
8. 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.
9. Matrix Methods for Linear Systems.
Review 1: Linear Algebraic Equations.
Review 2: Matrices and Vectors.
Linear Systems in Normal Form.
Homogeneous Linear Systems with Constant Coefficients.
Nonhomogeneous Linear Systems.
The Matrix Exponential Function.
10. Partial Differential Equations.
Introduction: A Model for Heat Flow.
Method of Separation of Variables.
Fourier Cosine and Sine Series.
The Heat Equation.
The Wave Equation.
11. Eigenvalue Problems and Sturm-Liouville Equations.
Introduction: Heat Flow in a Nonuniform Wire.
Eigenvalues and Eigenfunctions.
Regular Sturm-Liouville Boundary Value Problems.
Nonhomogeneous Boundary Value Problems and the Fredholm Alternative.
Solution by Eigenfunction Expansion.
Singular Sturm-Liouville Boundary Value Problems.
Oscillation and Comparison Theory.
12. Stability of Autonomous Systems.
Introduction: Competing Species.
Linear Systems in the Plane.
Almost Linear Systems.
Lyapunov's Direct Method.
Limit Cycles and Periodic Solutions.
Stability of Higher-Dimensional Systems.
13. Existence and Uniqueness Theory.
Introduction: Successive Approximations.
Picard's Existence and Uniqueness Theorem.
Existence of Solutions of Linear Equations.
Continuous Dependence of Solutions.
Method of Least Squares.
Runge-Kutta Precedure for n Equations.
Answers to Odd-Numbered Problems.
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