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Series Solution Of Ordinary Differential Equations

Series solutions of ordinary differential equations are a powerful method for finding approximate solutions to differential equations, especially when closed-form solutions are not readily obtainable. By expressing the solution as a power series, typically around a regular singular point, one can systematically derive coefficients through recursive relations or other techniques such as Frobenius method. This approach is particularly useful in cases where explicit solutions are challenging to obtain analytically, such as in nonlinear or higher-order differential equations. Series solutions enable a deeper understanding of the behavior of the system and provide a means to approximate solutions with desired accuracy. This method finds wide application in various fields including physics, engineering, and applied mathematics, contributing to the understanding and analysis of complex dynamical systems.

ABSTRACT

An ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE

Some ODEs may be solved explicitly in terms of known functions and integrals. When it is not possible, one may often use the equation for computing the Taylor series of the solutions. For applied problems, one generally uses numerical methods for ordinary differential equations for getting an approximation of the desired solution.

CHAPTER ONE

1.1 INTRODUCTION

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. In mathematics, there are three different types of differential equation which are:

i. Ordinary differential equations

ii. Partial differential equations

iii. Non-linear differential equations

However, in this work an ordinary differential equation (ODE) is studied which is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term “ordinary” is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

1.2 BACKGROUND OF THE STUDY

Ordinary differential equations (ODEs) arise in many contexts of mathematics and science. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related to each other via equations, and thus a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.

Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modelling), chemistry (reaction rates),^[2] biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).

Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d’Alembert, and Euler.

1.3 OBJECTIVE OF THE STUDY

The objective of this work is have a series solution to ordinary differential equation with different formulas.

1.4 APPLICATIONS OF THE STUDY

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

1.5 SCOPE OF THE STUDY

In this introductory course on Ordinary Differential Equations, we first provide basic terminologies on the theory of differential equations and then proceed to methods of solving various types of ordinary differential equations. We handle first order differential equations and then second order linear differential equations. We also discuss some related concrete mathematical modeling problems, which can be handled by the methods introduced in this course.

1.6 PURPOSE OF THE STUDY

This course is recommended for undergraduate students in mathematics, physics, engineering and the social sciences who want to learn basic concepts and ideas of ordinary differential equations. Learners are required to know usual college level calculus including differential and integral calculus.