Generally, students enrolled in Elementary Differential Equations courses are poorly prepared for rigorous treatment of the subject. I tried to alleviate this problem by isolating the material that requires greater sophistication than that normally acquired in the first year of calculus. The emphasis throughout is on making the work text readable by frequent examples and by including enough steps in working problems so that students will not be bogged down with complicated calculations.
This worktext has been written with the following objectives:
1. To provide in an elementary manner a reasonable understanding of differential equations for students of engineering and students of mathematics who are interested in applying their fields. Illustrative examples and practice problems are used throughout to help facilitate understanding. Whatever possible, stress is on motivation rather than following rules.
2. To demonstrate how differential equations can be useful in solving many types of problems – in particular, to show students how to: (a) translate problems into the language of differential equations, i.e. set up mathematical formulations of problems; (b) solve the resulting differential equations subject to given conditions; (c) interpret the solutions obtained.
3. To separate the theory of differential equations from their applications so as to give ample attention to each. This is accomplished by threatening theory and applications in separate lessons, particularly in early lessons of the coursebook. This is done for two reasons; First, from a pedagogical viewpoint, it seems inadvisable to mix theory and applications at an early stage since the students usually find applied problems difficult to formulate mathematically, and when they are forced to do this in addition to learning techniques for solution, it generally turns out that they learned neither effectively. By treating theory without applications and then gradually broadening out to applications (at the same time reviewing theory) the students may better master both since their attention is thereby focused only in one thing at a time. A second reason for separating theory and applications is enable instructors who may wish to present a minimum of applications to do so conveniently without being in the awkward position of having to skip around in lessons.
Inhaltsverzeichnis (Table of Contents)
- Preface
- Table of Contents
- Definition and Classification of Differential Equations
- Introduction
- Characteristics of Differential Equations
- Order
- Degree
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Non-Linear Differential Equations
- Eliminations of Arbitrary Constant
- Familiies of Curve
- Equations of Order One
- The Isoclines Of Equation
- The Existence Theorem
- Separtion of Variables
- Homogenous Functions
- Equation with Homogenous Coefficients
- Exact Equations
- !!!Linear Equation of Order One
- Integrating Factors Found by Inspection
- The Determination of Integrating Factors
- Bernoulli's Equation
- Applications of First Order Differential Equations
- Exponential Growth and Decay
- Newton's Law of Cooling
- Chemical Solutions
- Simple Electrical Circuits
- Homogeneous Linear Differential Equations with Constant Coefficient
- The Auxiliary Equation: Distinct Roots
- The Auxiliary Equation: Repeated Roots
- The Auxiliary Equation: Imaginary Roots
- Nonhomogeneous Equations: Undetermined Coefficient
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This worktext aims to provide a comprehensive understanding of differential equations for students of engineering and mathematics. It emphasizes readability through numerous examples and detailed problem-solving steps, focusing on motivation rather than strict rules.
- Understanding differential equations in an elementary manner.
- Demonstrating the practical applications of differential equations in various fields.
- Teaching students how to translate real-world problems into mathematical formulations using differential equations.
- Separating the theoretical aspects of differential equations from their applications.
- Providing a pedagogical approach that gradually introduces applications while reinforcing theoretical concepts.
Zusammenfassung der Kapitel (Chapter Summaries)
- Definition and Classification of Differential Equations: This chapter introduces the concept of differential equations and their classification based on order, degree, linearity, and type (ordinary or partial). It provides illustrative examples and discusses the historical context of their development.
- Characteristics of Differential Equations: This chapter explores the key features of differential equations, including order, degree, linearity, and the distinction between ordinary and partial differential equations. It delves into the significance of these characteristics in understanding and solving differential equations.
- Eliminations of Arbitrary Constant: This chapter focuses on techniques for eliminating arbitrary constants from equations. It provides step-by-step examples to illustrate the process of deriving differential equations from given functions.
- Familiies of Curve: This chapter examines families of curves, illustrating how differential equations can be used to represent and analyze these curves. It explores the relationship between differential equations and the properties of families of curves.
- Equations of Order One: This chapter delves into first-order differential equations, covering various methods for solving them, including separation of variables, homogeneous equations, exact equations, and integrating factors. It emphasizes practical applications of these techniques.
- The Existence Theorem: This chapter introduces the fundamental existence theorem for differential equations, providing theoretical underpinnings for the solvability of differential equations.
- Separtion of Variables: This chapter focuses on the method of separation of variables, demonstrating how to solve differential equations by separating the variables and integrating. It provides numerous examples to illustrate the technique.
- Homogenous Functions: This chapter defines homogenous functions and explores their role in solving differential equations. It introduces techniques for identifying and solving equations with homogenous coefficients.
- Exact Equations: This chapter examines exact differential equations and introduces methods for solving them. It provides examples to illustrate the concept of exactness and the steps involved in solving exact equations.
- !!!Linear Equation of Order One: This chapter covers linear first-order differential equations, highlighting their importance in various applications. It introduces methods for solving these equations, including integrating factors and other techniques.
- Integrating Factors Found by Inspection: This chapter explores methods for finding integrating factors by inspection, providing a practical approach to solving first-order differential equations.
- The Determination of Integrating Factors: This chapter presents a systematic method for determining integrating factors, enabling students to solve a wider range of first-order differential equations.
- Bernoulli's Equation: This chapter focuses on Bernoulli's equation, a special type of first-order differential equation. It introduces a transformation technique for solving Bernoulli's equations.
- Applications of First Order Differential Equations: This chapter showcases real-world applications of first-order differential equations, including models for exponential growth and decay, Newton's law of cooling, chemical reactions, and electrical circuits.
- Homogeneous Linear Differential Equations with Constant Coefficient: This chapter delves into homogeneous linear differential equations with constant coefficients, presenting methods for solving them using the auxiliary equation. It explores cases with distinct roots, repeated roots, and imaginary roots.
- Nonhomogeneous Equations: Undetermined Coefficient: This chapter introduces methods for solving nonhomogeneous linear differential equations using the method of undetermined coefficients. It provides practical examples to illustrate the technique.
Schlüsselwörter (Keywords)
This worktext delves into the realm of differential equations, covering key topics such as order, degree, linearity, ordinary and partial differential equations, methods of solution including separation of variables, homogeneous equations, exact equations, integrating factors, and the method of undetermined coefficients. It emphasizes applications in various fields like engineering, mathematics, and other disciplines.
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- Alan Nebrida (Author), 2022, Differential Equations. A Workbook, Munich, GRIN Verlag, https://www.grin.com/document/1277813
