Analysis of Differential Equations: Types, Solutions, and Uses

This essay provides an overview of differential equations, defining them as equations involving unknown functions and their derivatives, initially developed by Newton and Leibniz. It classifies differential equations into ordinary and partial, linear and non-linear, and homogeneous and non-homogeneous types, further categorizing partial differential equations into elliptic, parabolic, and hyperbolic. The essay discusses solution approaches, distinguishing between explicit formulas for simple equations and qualitative and numerical techniques for complex, non-linear equations. It highlights the broad applications of differential equations in modeling dynamic systems in biology, finance, physics, and chemistry, including heat and sound propagation, fluid flow, stock market prediction, elasticity, electrodynamics, and electrostatics. The document concludes with a list of references.

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Running head: DIFFERENTIAL EQUATIONS
Title: Differential Equations
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Differential Equations
Introduction
A differential equation can be defined as an equation encompassing an unknown function
and its derivatives. Differential equations (DE) were first developed by Isaac Newton and
Leibniz in 1695 and have since been advanced by mathematicians over the years (Hale and
Lunel ,2013). DEs are are identified by their order and degree. The order of differential equation
is equal to the highest power to which the derivatives are raised while its degree is order of the
highest derivative. Differential equations is abroad area of study and is applied in fields of pure
mathematics, applied mathematics, physics, and engineering and biology fields. The list is
however non-exhaustive. The fields utilize the different properties of numerous differential
equations.
Types of differential equations
Differential equations are classified into several types based on their properties. The type
of DE is useful in deciding the approach to a problem’s solution. Most general classes of
classification are ordinary and partial, linear and non-linear and homogeneous and non-
homogenous differential equations (Perko ,2013). .
The distinction between ordinary and partial differential equation is the number of
variables and their derivatives. Ordinary differential contains one variable while partial
differential equations contain more than one variable and their respective derivatives. Partial
differential equations may further be classified into elliptic, parabolic and hyperbolic equations
(Evans ,2010).
A differential equation is said to be linear if the variables and their derivatives are in
linear form. On the other hand, non-linear equations are is a product of unknown function and
derivatives whose degree is greater than one.
Solution
Pure mathematics insists of specific solutions to a differential problems wile applied
mathematics focuses on approximating the solutions to the problems .Simple differential
equations can be solved explicitly through representation of formulas. However, different

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Differential Equations
methods such as qualitative and numerical techniques are used to solve complex equations
especially non-linear equations (Hirsch, Smale and Devaney,2012). .
Applications of Equations
Differential equations are useful in describing and solving any unknown phenomena that
changes with time. In biology and finance, differential equations are used in modelling activities
of intricate systems. Numerous physics and chemistry laws can be described using differential
equations. Some of the problems that are solved using differential equations include heat and
sound propagation, fluid flow, predicting stock market prices, elasticity, electrodynamics
electrostatics just to name a few.

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Differential Equations
References
Hale, J. K., & Lunel, S. M. V. (2013). Introduction to functional differential equations (Vol. 99).
Springer Science & Business Media.
Perko, L. (2013). Differential equations and dynamical systems (Vol. 7). Springer Science &
Business Media.
Evans, L. C. (2010). Partial differential equations.
Hirsch, M. W., Smale, S., & Devaney, R. L. (2012). Differential equations, dynamical systems,
and an introduction to chaos. Academic press.