1.1.Introduction
¬¬¬
Differential equations arise in many areas of science and technology; whenever a deterministic relationship involving some continuously changing quantities modeled by functions) and their rates of change (expressed as derivatives)is known or postulated. This is illustrated by classical mechanics, where the motion of a body is described by its position, velocity, acceleration and various forces ting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In many cases, this differential equation may be solved, yielding the law of motion. Canonical forms are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many 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 curacy. Differential equations play a prominent role in engineering, physics, economics and other disciplines. The study of differential equations is a wide field in both pure and applied mathematics.

The subject of differential equations constitutes a large and very important branch of modern Mathematics. The study of differential equations is important because they x frequently used in application of Mathematics to problems in Science, e.g. velocity or acceleration in the study of motion generally supplies us with a differential equations, satisfied by unknown function. Differential equations play a prominent role engineering, physics,...

...
STABILITY OF SOLUTIONS OF NON-LINEAR ORDINARY DIFFERENTIALEQUATIONS.
CERTIFICATION
This is to certify, that this project work title “STABILITY OF SOLUTIONS OF NON-LINEAR ORDINARY DIFFERENTIALEQUATIONS” submitted to the Department of Mathematics, College of Natural and Applied Science, Michael Okpara University of agriculture Umudike. For the award of Bachelor of Science (B.Sc.) degree in Mathematics is research work carried out by Ukazim Great Kelechi with registration number MOUAU/08/12869
Ukazim Great Kelechi ---------------------------
Signature/date
Mr. Emenogu G.N. ---------------------------
(Supervisor) Signature/date
Dr. OGBU H.M ---------------------------
(Head of Department) Signature/date
Prof O.K. Achi
(Dean CNAS) ---------------------------
Signature/date
----------------------------- ------------------------------
(External Supervisor) Signature/date
DEDICATION
I dedicate this work to God Almighty the giver of wisdom and to the loving memory of my Grandfather.
ACKNOWLEDGMENT
Without much ado and total humility, I wish to appreciate God Almighty for his mercies and favour on my life and making it possible for this work to be accomplished. In the same manner, I appreciate the effort of my Supervisor Mr. Emelogu...

...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...

...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 ofdifferentialequations 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 =...

...Heat Equation from PartialDifferentialEquations An Introduction (Strauss)
These notes were written based on a number of courses I taught over the years in the U.S.,
Greece and the U.K. They form the core material for an undergraduate course on Markov
chains in discrete time. There are, of course, dozens of good books on the topic. The
only new thing here is that I give emphasis to probabilistic methods as soon as possible.
Also, I introduce stationarity before even talking about state classification. I tried to make
everything as rigorous as possible while maintaining each step as accessible as possible. The
notes should be readable by someone who has taken a course in introductory (non-measuretheoretic) probability.
The first part is about Markov chains and some applications. The second one is specifically
for simple random walks. Of course, one can argue that random walk calculations should
be done before the student is exposed to the Markov chain theory. I have tried both and
prefer the current ordering. At the end, I have a little mathematical appendix.
There notes are still incomplete. I plan to add a few more sections:
– On algorithms and simulation
– On criteria for positive recurrence
– On doing stuff with matrices
– On finance applications
– On a few more delicate computations for simple random walks
– Reshape the appendix
– Add more examples
These are things to...

...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...

...6 Systems Represented by Differential and Difference Equations
Recommended Problems
P6.1 Suppose that y 1(t) and y 2(t) both satisfy the homogeneous linear constant-coeffi cient differentialequation (LCCDE)
dy(t) + ay(t) = 0
dt
Show that y 3 (t) = ayi(t) + 3y2 (t), where a and # are any two constants, is also a solution to the homogeneous LCCDE. P6.2 In this problem, we consider the homogeneous LCCDE d 2yt + 3 dy(t) + 2y(t) = 0
dt 2 dt
(P6.2-1)
(a) Assume that a solution to eq. (P6.2-1) is of the form y(t) = es'. Find the qua dratic equation that s must satisfy, and solve for the possible values of s. (b) Find an expression for the family of signals y(t) that will satisfy eq. (P6.2-1). P6.3 Consider the LCCDE
dy(t) + 1 y(t) = x(t), 2 dt x(t) = e- t u(t) (P6.3-1)
(a) Determine the family of signals y(t) that satisfies the associated homogeneous equation. (b) Assume that for t > 0, one solution of eq. (P6.3-1), with x(t) as specified, is of the form
y 1(t) = Ae-, t > 0
Determine the value of A. (c) By substituting into eq. (P6.3-1), show that
y 1(t) = [2e -t/2 - 2e-']u(t)
is one solution for all t.
P6-i
Signals and Systems
P6-2
P6.4
Consider the block diagram relating the two signals x[n] and y[n] given in Figure P6.4.
x[n]
+
1 y[n]
1 2...

...MATHEMATICAL METHODS
PARTIALDIFFERENTIALEQUATIONS
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...

...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...

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