Page 133 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II Sachin Kaushal, Lovely Professional University

Notes Unit 10: Homogeneous Equations

CONTENTS
Objectives
Introduction

10.1 Homogeneous Equations
10.2 Equations Reducible to Homogeneous Form
10.3 Summary

10.4 Keywords
10.5 Review Questions
10.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the concept of homogeneous equations
 Discuss the equations reducible to homogeneous form

Introduction

As we know, the finest manner to resolve a new problem is to lessen it, in some manner, into the
outline of a problem that you already recognize how to solve. This is what you perform with
homogeneous differential equations. If you identify the truth that an equation is homogeneous
you can, in some cases, carry out a substitution which will permit you to apply separation of
variables to solve the equation. At this position you may be inquiring yourself, what is a
homogeneous differential equation? It is just an equation where both coefficients of the
differentials dx and dy are homogeneous.

10.1 Homogeneous Equations

Homogeneous functions are defined as functions where the sums of the powers of each term are
the same. A homogeneous equation can be malformed into a distinguishable equation by a
change of variables.

An equation of the form
dy f 1  ,x y 
 …..(1)
dx f 2  , x y 

is called a homogeneous function of the same degree in x and y.
If f (x, y) and f (x, y) are homogeneous functions of degree n in x and y, then
1 2
y
y
f 1  , x y   x n  1   and f 2  , x y   x n  2  
 
 
x
x
   
 Equation (1) reduces to

128 LOVELY PROFESSIONAL UNIVERSITY