This book is aimed to undergraduate STEM majors and to researchers using ordinary differential equations. It covers a wide range of STEM-oriented differential equation problems that can be solved using computational power series methods. Many examples are illustrated with figures and each chapter ends with discovery/research questions most of which are accessible to undergraduate students, and almost all of which may be extended to graduate level research. Methodologies implemented may also be useful for researchers to solve their differential equations analytically or numerically. The textbook can be used as supplementary for undergraduate coursework, graduate research, and for independent study.
Author(s): James Sochacki, Anthony Tongen
Series: Problem Books in Mathematics
Publisher: Springer
Year: 2023
Language: English
Pages: 219
City: Cham
Tags: Differential Equations; Power Series; Dynamical Systems; Numerical Solutions; Polynomials; STEM; Numerical Approximation; Ordinary Differential Equations; ODE; Linear ODE; Poincaré-Bendixson Theory; Newton's Method; Newton's N-Body Problem; Maclaurin Polynomials
Preface
Acknowledgment
Contents
1 Introduction
1.1 Linear Differential Equations
1.2 Power Series Solutions
1.3 Picard Iterates
1.4 A Second Order Linear Differential Equation
1.5 The Independent Variable t
Exercises
References
2 Egg 1: The Quadratic Ordinary Differential Equation
2.1 The Quadratic ODE
2.2 Questions, Projects and Future Consideration
Exercises
References
3 Egg 2: A First Order Differential Equation with Exponent
Exercises
References
4 Egg 3: The First Order Sine Differential Equation
Exercises
References
5 Egg 4: A Second Order ODE with Exponent
Exercises
Reference
6 Egg 5: The Second Order Sine ODE—The Single Pendulum
6.1 The Lagrangian Derivation
Exercises
References
7 Egg 6: Newton's Method and the Steepest Descent Method
7.1 Newton's Method
7.2 The Method of Steepest Descent
7.3 Systems of Nonlinear Equations
Exercises
References
8 Egg 7: Determining Power Series for Functions Through ODEs
8.1 Inverse of a Function
8.2 Functions with Singularities
8.3 Rational Power Series
Exercises
References
9 Egg 8: The Periodic Planar ODE
Exercises
References
10 Egg 9: The Complex Planar Quadratic ODE
Exercises
Reference
11 Egg 10: Newton's N-Body Problem
Exercises
References
12 Egg 11: ODEs and Conservation Laws
Exercises
References
13 Egg 12: Delay Differential Equations
Exercises
References
14 An Overview of Polynomial ODEs
14.1 First Order Quadratic ODEs
14.2 General Quadratic ODEs
Exercises
References
A A Review of Maclaurin Polynomials and Power Series
Exercises
B The Dog Rabbit Chasing Problem
Exercises
References
C A PDE Example: Burgers' Equation
Exercises
References
References
