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 Constant
Sir Isaac Newton (December 25, 1642 – March 20, 1727) was an English
physicist and mathematician who is widely regarded as one of the most influential
scientists of all time and as a key figure in the scientific revolution. His book
Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural
Philosophy"), first published in 1687, laid the foundations for most of classical
mechanics. Newton also made seminal contributions to optics and shares credit with
Gottfried Leibniz for the invention of the infinitesimal calculus.
In 1671, Newton wrote his then-unpublished The Method of Fluxions and
Infinite Series (published in 1736), in which he classified first order differentialequations, known to him as fluxional equations, into three classes. The first two
classes contain only ordinary derivatives of one or more dependent variables, with
respect to a single independent variable, and are known today as "ordinary...

...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 growth or decay, for example. In this chapter we study some other types of first-order
differentialequations. They involve only first derivatives of the unknown function.
15.1
Solutions, Slope Fields, and Picard’s Theorem
We begin this section by defining general differentialequations involving first derivatives.
We then look at slope fields, which give a geometric picture of the solutions to such equations. Finally we present Picard’s Theorem, which gives conditions under which first-order
differentialequations have exactly one solution.
General First-Order DifferentialEquations and Solutions
A first-order differentialequation is an equation
dy
= ƒsx, yd
dx
(1)
in which ƒ(x, y) is a function of two variables defined on a region in the xy-plane. The
equation is of first order...

...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 differentialequations of first order, namely, the class of differentialequations of first order where variables x and y can be separated, the class of exact equations (equation (2.3) is to be satisfied by the coefficients of dx and dy, the class of linear differentialequations having a standard form (2.7) and the class of those differentialequations of first order which can be reduced to separable differentialequations or standard linear form by appropriate.
2.1 Separable Variables
Definition 2.1: A first order differentialequation of the form
[pic]
is called separable or to have separable variables.
Method or Procedure for solving separable differentialequations
(i) If h(y) = 1, then
[pic]
or dy = g(x) dx
Integrating both sides we get
[pic]
or [pic]...

...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 ride back and forth to school, or what really happens to an object as it falls through the air, even how that people can predict an approximate future population. These questions, and many more, can be answered and explained through different variations of differentialequations. By explaining and answering even just one of these questions through different differentialequations I will also be answering two other important questions. Why is differentialequations required for many students and how does it apply in the career of a mechanical engineer?
First some background. What is a differentialequation?
A differentialequation is a mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. They are used whenever a rate of change is...

...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
– Echelon form, Normal form – Solution of Linear Systems – Direct Methods – LU
Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution
of Linear Systems.
Eigen values, Eigen vectors – properties – Condition number of Matrix, Cayley –
Hamilton Theorem (without proof) – Inverse and powers of a matrix by Cayley –
Hamilton theorem – Diagonalization of matrix – Calculation of powers of matrix –
Model and spectral matrices.
Real Matrices, Symmetric, skew symmetric, Orthogonal, Linear Transformation -
Unit-III
Linear
Transformations
Orthogonal Transformation. Complex Matrices, Hermition and skew Hermition
matrices, Unitary Matrices - Eigen values and Eigen vectors of complex matrices and
their properties. Quadratic forms - Reduction of quadratic form to canonical form,
Rank, Positive, negative and semi definite, Index, signature, Sylvester law, Singular
value decomposition.
Solution of Algebraic and Transcendental Equations-...

...determine the bulk 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 Equation 2.
(Eqn. 2)
Plugging in known values to Equation 2, we find:
The experimental friction factor of the pipe can be calculated as:
(Eqn. 3)
Using the pressure drop for the chosen sample from smallest smooth copper pipe across the known distance L, we obtain an experimental friction factor
The theoretical friction factor for smooth pipes can be calculated with the Petukhov formula:
(Petukhov Formula)
Using this formula with our calculated Reynolds number yields a theoretical friction factor of
Because Pipe 4 is a rough pipe, this Petukhov Formula does not apply and we must perform additional sample calculations. From the first data point for the fourth pipe we obtain the following flow properties:
Using Equations 2 and 3 we can find the following Reynolds number and experimental friction factor:
The theoretical friction factor for a rough pipe can be found by calculating the parallel...

...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 that we try to understand
in our technological era. They can be used to describe the way
planets revolve around the sun, the way rockets travel to outer space, the
way chemicals interact, the way electricity flows, even the way matter itself
exists. They can also be used to describe economic processes, such as the
way the net worth of a company changes. We will only be able to scratch
the surface of this very important subject.
Recall that a differentialequation (d.e. for short) is any equation that involves
at least one derivative (of an unknown function). Solving a d.e. means
finding (all) functions that satisfy the d.e.
Here are two examples:
1.
dy
dx
= cos x
2.
d2p
dt2
− 4
dp
dt
+ 3p = 0
In Example 1, the independent variable is x; one then tries to find a formula
describing how y depends on x. We call y the dependent variable. In
Example 2, the independent variable is t, the dependent variable is p.
The highest order...

...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 http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
Preface
What follows are my lecture notes for Math 2351/2352: Introduction to ordinary differentialequations/Differentialequations and applications, taught at the Hong Kong University of Science and Technology. Math 2351, with two lecture hours per week, is primarily for non-mathematics majors and is required by several engineering and science departments; Math 2352, with three lecture hours per week, is primarily for mathematics majors and is required for applied mathematics students. Included in these notes are links to short tutorial videos posted on YouTube. There are also some links to longer videos of in-class lectures. It is hoped that future editions of these notes will more seamlessly join the video tutorials with the text. Much of the material of Chapters 2-6 and 8...

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