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SERIES SOLUTION OF ORDINARY
DIFFERENTIAL EQUATION
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 (social as well as natural). 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.
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