Homogeneous Equation

Homogeneous Equation:

Introduction to Homogeneous Equation:


Homogeneous Equation:

The equation is said to be homogeneous when the function of the equation is homogeneous.The common question that is very frequently asked is,

Ques 1.)How to write a Homogeneous Equation????Below is the given answer to this question:

Ans) The homogeneous equation can be written in the general form that is homogeneous function on one side of the equal to symbol and zero on the other side.

The general form is

(Homogeneous function) F(X) =0

The function Is same for all S, then

F (Sx, Sy) =Sn F(x, y)

Properties of Homogeneous Equation:

Property: 1

dy/dx=f(x, y) =g(y/x)

If the condition gets satisfied, then the equation P(x, y) dx+Q(x, y) dy=0 (differential form) is called as homogeneous. Here “g”is a function.

Practice problem:

Problem 1: find the whether the given equation is homogeneous or not? The equation is (x2 + 3y2) dx – xy dy = 0.

Solution:

(x2 + 3y2) dx = xy dy.

dy/dx=(x2+3y2)/ xy.

=(x2/xy) + (3y2/xy).

= x/y+3y2/xy

=x/y+3y/x

= x/y +3y/x

= (y/x)-1+3(y/x) =g(y/x).

Property 2:

A function F (p, q) of the variables p and q is said to be homogeneous of degree n, now consider a parameter S

F (sp, sq) = S2 F (p, q)

Practice problem:

Problem:

F(p,q) = p2 –4p2q + q3

it is a homogeneous function of degree 3

Since

F (sp, sq) = (sp) 3 – 4(sp) 2(sq) + (sq) 3

= s3 (p3– 4p2q + q3)

Let us now solve few problems related to Homogeneous Function:

Examples:

Find whether the given equation is homogeneous or not?

Problem: 1

F(x,y)= 3x3+ 4xy +12y3

Solution:

Here, the degree of x is 3+1= 4

The degree of y is 1+3=4

So the degree of both x and y are equal

Answer:The given equation is homogeneous equation.

Problem: 2

F(x,y)= 3x3+ 4x2y +12y3

Solution:

Here the degree of x is 3+2=5

The degree of y is 1+3=4

So the degree of both x and y are not equal

Answer:The given equation is not homogeneous equation.

Problem: 3

F(x,y)= 3x3+ 16x2 y2 +12y3

Solution:

Here the degree of x is 3+2= 5

The degree of y is 2+3=5

So the degree of both x and y are equal

Answer:The given equation is homogeneous equation.


Hope you like the above example of Homogeneous Equation.Please leave your comments, if you have any doubts.