Without knowing something about differential equations and methods of solving them, it is difficult to appreciate the history of this important branch of mathematics. Further, the development of differential equations is intimately interwoven with the general development of mathematics and cannot be separated from it. Nevertheless, to provide some historical perspective, we indicate here some of the major trends in the history of the subject, and identify the most prominent early contributors. Other historical infor- mation is contained in footnotes scattered throughout the book and in the references listed at the end of the chapter. The subject of differential equations originated in the study of calculus by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) in the seventeenth century. Newton grew up in the English countryside, was educated at Trinity College, Cambridge, and became Lucasian Professor of Mathematics there in 1669. His epochal discoveries of calculus and of the fundamental laws of mechanics date from 1665. They were circulated privately among his friends, but Newton was extremely sensitive to criticism, and did not begin to publish his results until 1687 with the appearance of his most famous book, Philosophiae Naturalis Principia Mathematica. While Newton did relatively little work in differential equations as such, his development of the calculus and elucidation of the basic principles of mechanics provided a basis for their applications in the eighteenth century, most notably by Euler. Newton classified first order differential equations according to the forms dy/dx = f (x), dy/dx = f (y), and dy/dx = f (x,y). For the latter equation he developed a method of solution using infinite series when f (x,y) is a polynomial in x and y. Newton’s active research in mathematics ended in the early 1690s except for the solution of occasional challenge problems and the revision and publication of results obtained much earlier. He was...

...CHAPTER 1
INTRODUCTION TO
DIFFERENTIALEQUATIONS
1
Chapter INTRODUCTION TO
DIFFERENTIAL
1
EQUATIONS
Outline:
1.1
1.2
1.3
1.4
1.5
1.6
Basic Definition
Types of DifferentialEquations
Order of a DifferentialEquation
Degree of a DifferentialEquation
Types of Solutions to a DifferentialEquation
Elimination of Arbitrary...

...FIRST-ORDER
DIFFERENTIALEQUATIONS
OVERVIEW In Section 4.8 we introduced differentialequations of the form dy>dx = ƒ(x),
where ƒ is given and y is an unknown function of x. When ƒ is continuous over some interval, we found the general solution y(x) by integration, y = 1 ƒ(x) dx. In Section 6.5 we
solved separable differentialequations. Such equations arise when investigating exponential...

...CHAPTER 2
FIRST ORDER DIFFERENTIALEQUATIONS
2.1 Separable Variables
2.2 Exact Equations
2.2.1 Equations Reducible to Exact Form.
2.3 Linear Equations
4. Solutions by Substitutions
2.4.1 Homogenous Equations
2.4.2 Bernoulli’s Equation
2.5 Exercises
In this chapter we describe procedures for solving 4 types of differential...

...Doctor Gary Hall
DifferentialEquations
March 2013
DifferentialEquations in Mechanical Engineering
Often times college students question the courses they are required to take and the relevance they have to their intended career. As engineers and scientists we are taught, and even “wired” in a way, to question things through-out our lives. We question the way things work, such as the way the shocks in our car work to give us a smooth...

...MATHEMATICAL METHODS
PARTIAL DIFFERENTIALEQUATIONS
I YEAR B.Tech
By
Mr. Y. Prabhaker Reddy
Asst. Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad.
SYLLABUS OF MATHEMATICAL METHODS (as per JNTU Hyderabad)
Name of the Unit
Unit-I
Solution of Linear
systems
Unit-II
Eigen values and
Eigen vectors
Name of the Topic
Matrices and Linear system of equations: Elementary row transformations – Rank
–...

...velocity of the stream using Equation 1.
(Eqn. 1)
Where is the flowrate in m3/s and A is the cross-sectional area of the pipe. To find the flowrate, we multiply the flowmeter reading by the constant
and convert from gallons to cubic meters as follows:
The cross sectional area of the 7.75mm pipe is
Plugging these values into Equation 1, we obtain a bulk velocity .
With the bulk velocity value, we can find the Reynolds number of the flow using...

...Differentialequations
1 Introduction
These notes are to be read together with Chapter 7 in the textbook (Calculus:
Concepts and contexts, by James Stewart). Separable differentialequations
are dealt with in the textbook and in these notes; the notes then continue
with first order linear differentialequations.
Differentialequations describe most, if not all, processes...

...Introduction to DifferentialEquations
Lecture notes for MATH 2351/2352 (formerly MATH 150/151)
Jeffrey R. Chasnov
The Hong Kong University of Science and Technology
The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong
c Copyright ○ 2009–2012 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit...

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