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"A First Course in Differential Equations with Modeling Applications," 12th Edition by Dennis G. Zill, offers a comprehensive introduction to differential equations, balancing analytical, qualitative, and quantitative approaches. Below is a chapter-wise summary:
1. Introduction to Differential Equations
o Covers definitions, terminology, and the formulation of differential equations as mathematical models.
2. First-Order Differential Equations
o Discusses methods for solving first-order equations, including separable, exact, and linear equations, as well as applications.
3. Modeling with First-Order Differential Equations
o Explores real-world applications such as population dynamics, mixing problems, and Newton's law of cooling.
4. Higher-Order Differential Equations
o Examines linear differential equations of second and higher orders, focusing on homogeneous and nonhomogeneous cases.
5. Modeling with Higher-Order Differential Equations
o Applies higher-order equations to mechanical vibrations, electrical circuits, and beam deflection problems.
6. Series Solutions of Linear Equations
o Introduces power series methods for solving differential equations near ordinary and singular points.
7. The Laplace Transform
o Covers the Laplace transform technique for solving linear differential equations and systems, including step and impulse functions.
8. Systems of Linear First-Order Differential Equations
o Discusses methods for solving linear systems, including matrix approaches and eigenvalue analysis.
9. Numerical Solutions of Ordinary Differential Equations
o Introduces numerical methods such as Euler's method and Runge-Kutta methods for approximating solutions.
10. Plane Autonomous Systems
o Explores qualitative analysis of two-dimensional systems, including phase plane analysis and stability.
11. Partial Differential Equations and Fourier Series
o Provides an introduction to partial differential equations and solutions using Fourier series.
This edition is designed to engage students across various disciplines, offering numerous examples, clear explanations, and practical applications to illustrate the relevance of differential equations in modeling real-world phenomena.