CHAPTER 2
FIRST ORDER DIFFERENTIAL EQUATIONS
2.1 Separable Variables
2.2 Exact Equations
2.2.1Equations Reducible to Exact Form.
2.3 Linear Equations
4. Solutions by Substitutions
2.4.1Homogenous Equations
2.4.2 Bernoulli’s Equation
2.5 Exercises
In this chapter we describe procedures for solving 4 types of differential equations of first order, namely, the class of differential equations 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 differential equations having a standard form (2.7) and the class of those differential equations of first order which can be reduced to separable differential equations or standard linear form by appropriate. 2.1 Separable Variables

Definition 2.1: A first order differential equation of the form

[pic]
is called separable or to have separable variables.

Method or Procedure for solving separable differential equations

(i) If h(y) = 1, then
[pic]
ordy = g(x) dx
Integrating both sides we get
[pic]
or[pic]
where c is the constant of integral
We can write
[pic]
where G(x) is an anti-derivative (indefinite integral) of g(x) (ii) Let [pic]
where [pic],
that is f(x,y) can be written as the product of two functions, one function of variable x and other of y. Equation
[pic]
can be written as
[pic]
By integrating both sides we get
[pic]
where [pic]
or [pic]
where H(y) and G(x) are anti-derivatives of [pic]and [pic], respectively.

Example 2.1: Solve the differential equation

[pic]

Solution: Here [pic], [pic]and [pic]

[pic], [pic]

Hence

H(y) = G(x) + C

or lny = lnx + lnc(See Appendix )
lny – lnx = lnc
[pic]
[pic]
y = cx

...MAT114
Multivariable Calculus and DifferentialEquations
Version No. 1.00
Course Prerequisites
L T P C
3 0 2 4
: 10+2 level Mathematics/ Basic Mathematics (MAT001)
Objectives
This Mathematics course provides requisite and relevant background necessary to understand the
other important engineering mathematics courses offered for Engineers and Scientists. Three
important topics of applied mathematics, namely the Multiple integrals, Vector calculus, Laplace
transforms which require knowledge of integration are introduced.
Expected Outcome
At the end of this course the students are expected to learn
(i)
how to evaluate multiple integrals in Cartesian, Cylindrical and Spherical geometries.
(ii)
the powerful language of Vector calculus with physical understanding to deal with
subjects such as Fluid Dynamics and Electromagnetic fields.
(iii)
to solve ordinary differentialequations directly and also use transform methods where its
possible
Unit 1
Mutivariable Calculus
9L+4P hours
Functions of two variables-limits and continuity-partial derivatives –total differential–Taylor’s
expansion for two variables–maxima and minima–constrained maxima and minima-Lagrange’s
multiplier method- Jacobians
Unit 2
Mutiple Integrals
9L+4P hours
Evaluation of double integrals–change of order of integration– change of variables between
cartesian and polar co-ordinates- evaluation of triple integrals-change of variables...

...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...

...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-...

...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...

...Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary DifferentialEquations
1.0 Introduction
In mathematics, if y is a function of x, then an equation that involves x, y and one or more derivatives of y with respect to x is called an ordinary differentialequation (ODE). The ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary function in close form. In addition, the ODEs is use to solve many problems in real life such as cooling or warming law, radio-active decay, carbon dating and in social issue like predator-prey models and exponential growth model.
In this proposal, we are concerned with the numerical solution of initial value problem (IVP) with two fixed points for ODE. The general form is
y'=(y-v1)(y-v2)gy, (1)
given initial values yxn=yn, where v1<v2 ϵ R and g(y)≠0 is a bounded real-valued function with continuous derivatives. Assume the fixed points are y1x=0 and y2(x)=1.
Diagonally Implicit Two-point Block Backward Differentiation Formulas (DI2BBDF) is a new method from the continuation of the previous methods. One of the methods is block method which is used to compute k blocks and to calculate the current block where each block contain r points. The general form for r point k block is...

...Ordinary DifferentialEquations
[FDM 1023]
Chapter 1
Introduction to Ordinary
DifferentialEquations
Chapter 1:
Introduction to DifferentialEquations
Overview
1.1. Definitions
1.2. Classification of Solutions
1.3. Initial and Boundary Value Problems
1.1. Definitions
Learning Outcomes
At the end of the section, you should be able to:
1) Define a differentialequation
2) Classify differentialequations by type, order
and linearity
Recall
Dependent and Independent variables?
Example 1
Consider an equation y = 4 x + 1
The values of y depend on the choice of x.
y - dependent variable (function)
x - independent variable (argument)
Example 2
For a parametric equations of the unit circle
The values of y and x depend on the choice of θ.
y and x - dependent variable
θ - independent variable
What is
represents?
It represents the derivative of dependent
variable y with respect to independent
variable x.
By first principle, it is defined as rate of
change of y with respect to x.
1.1. Definitions
What is a DifferentialEquation?
A differentialequation (DE) is an
equation containing the derivatives of one
or more dependent variables with respect
to one or more independent...

...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...

...Without knowing something about differentialequations and methods of solving them, it is difficult to appreciate the history of this important branch of mathematics. Further, the development of differentialequations 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 differentialequations 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 differentialequations as such, his...