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...

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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...

...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
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...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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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...

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Chapter 1
Introduction to Ordinary
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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:
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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
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Differentialequations describe most, if not all, processes...

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