To: Team MARKsmen of ENGL301 99c
From: Ayazhan Makhambetova, Team MARKsmen member
Date: January 23, 2019
Subject: Revised Definition Assignment (Assignment 1:3)
Dear MARKsmen,
As stated on the English 301 course blog, the purpose of this assignment is to choose a relatively complex term related to student’s area of studies and give it a definition using three different definition categories: parenthetical, sentence and extended. I have noticed that we all are coming from scientific backgrounds, therefore I have chosen a relatively complex mathematical term: differential equation.
- Parenthetical definition:
Differential equation (an equation that contains derivatives)
- Sentence definition:
Differential equation explains how a rate of change (‘derivative’) in one variable is related to other variables.
- Expanded definition:
History & Etymology:
The history of differential equations begins with a mathematician named Gottfried Wilhelm von Leibniz, who in 1675 first wrote a differential equation and later, in 1682, published a first paper on this topic. His article has gained attention from a lot of famous mathematicians and physicists of those times, including James and John Bernoulli. Brothers Bernoulli have made a great impact into this new field of mathematical analysis, bringing new discoveries and methods of problem solving.
Another notable scientist, who worked with differential equations would be Isaac Newton. He has classified differential equations into multiple classes, focusing on the first order equation (i.e. first derivative).
Importance and General purpose:
First of all, differential equations help mathematician better understand the functions.
They arise in the studies of rates of change and of quantities of things that change. Many fundamental laws of physics and chemistry can be written in a differential equation form. Differential equations have a significant role in virtually modelling physical, technical, or even biological processes. They are used in biology and economics to model the behaviour of complex systems.
Operating principle & Analysis of Parts:
The simplest type of differential equation, an ordinary differential equation, contains an unknown function of one variable x, it’s derivatives and some other given functions of x.
Additional clarification on derivatives:
In mathematics, derivative of a function shows the rate of change, i.e. the amount by which a function is changing at some given point. It is written using “dy over dx” notation, which means “the difference in y divided by the difference in x”.
Required conditions:
Since we are dealing with derivatives and anti-derivatives, the given functions should be continuous, meaning that the function should not have any holes in it. A “hole” in the function would be the area where the function fails, for example gets divided by zero (forbidden operation), or falls below the scope of number it covers.
Example:
One of the most famous differential equations is Newton’s Second Law of Motion.
Where: F is force acted on the given object, m is that object’s mass, and a is object’s acceleration.
At the first glance, it might not appear as a differential equation. However, if we rewrite it in more complicated way, using objects velocity v (note that velocities first derivative is acceleration), time t and the position of an object u. It should result in:
Now, we can see the derivatives involved, which makes this a differential equation.
Works Cited:
Dawkins, P. Differential Equations. Paul’s Online Notes, 2018, http://tutorial.math.lamar.edu/Classes/DE/DE.aspx
Samko, S G, A A. Kilbas, and O I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Switzerland: Gordon and Breach Science Publishers, 1993. Print.
Sasser, J. (2015) History of Ordinary Differential Equations: The First Hundred Years. MATHEMATICAL ORIGINS OF ORDINARY DIFFERENTIAL EQUATIONS: THE FIRST HUNDRED YEARS, http://www2.fiu.edu/~yuasun/ODE_History.pdf.
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