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# Homogeneous Functions and EulerтАЩs Theorem

These types of functions are important in Ordinary differential equations.

Homogeneous Functions and EulerтАЩs Theorem

Definition 8.12

(a) Let A = {( x, y ) | a < x < b, c < y < d } тКВ тДЭ2 , F : A тЖТ тДЭ, we say that F is a homogeneous function on A , if there exists a constant p such that F ( ╬╗ x, ╬╗ y ) = ╬╗ p F ( x , y) for all ╬╗ тИИ тДЭ and sutitably restricted ╬╗, x, y, such that (╬╗ x , ╬╗ y) тИИ A . This constant p is called degree of F .

(b) Let B = {( x, y , z) | a < x < b , c < y < d, u < z < v} тКВ тДЭ3 , G : B тЖТ тДЭ, we say that G is a homogeneous function on B , if there exists a constant p such that G ( ╬╗ x, ╬╗ y , ╬╗ z ) = ╬╗ pG ( x, y , z) for all ╬╗ тИИ тДЭ and sutitably restricted ╬╗, x, y, z, such that (╬╗ x , ╬╗ y, ╬╗z ) тИИ B . This constant p is called degree of G .

Note: Division by any variable may occur, to avoid division by zero, we say that ╬╗, x, y, z are sutitably restricted real numbers.

These types of functions are important in Ordinary differential equations (Chapter 10). Let us consider some examples.

Consider F ( x, y) = x3 тИТ 2 y3 + 5xy2 , (x , y) тИИ тДЭ2 . Then

F ( ╬╗ x , ╬╗ y) = (╬╗ x ) 3 тИТ 2(╬╗ y)3 + 5(╬╗ x )(╬╗ y)2 = ╬╗3 (x 3 тИТ 2 y3 + 5xy2 )

and hence F is a homogeneous function of degree 3.

On the other hand,

G ( x, y) = ex2 + 3y2 is not a homogeneous function

because, G ( ╬╗ x , ╬╗ y) = e (╬╗x)2  + 3(╬╗ y)2 тЙа ╬╗pG ( x, y)

for any ╬╗ тЙа 1 and any p.

Example 8.21

Show that is a homogeneous function of degree 1.

Solution

We compute for all ╬╗ тИИ тДЭ. So F is a homogeneous function of degree 1.

We state the following theorem of Leonard Euler on homogeneous functions.

Definition 8.13 (Euler)

Suppose that A = {( x, y ) | a < b, c < y < d }тКВ тДЭ2, F : A тЖТ тДЭ2 . If F is having continuous partial derivatives and homogeneous on A , with degree p , then Suppose that B = {( x, y , z) | a < x < b, c < y < d, u < z < v} тКВ тДЭ3 , F : B тЖТ тДЭ3. If F is having continuous partial derivatives and homogeneous on B , with degree p , then We omit the proof. The above theorem is also true for any homogeneous function of n variables; and is useful in certain calculations involving first order partial derivatives.

Example 8.22 Solution

Note that the function u is not homogeneous. So we cannot apply EulerтАЩs Theorem for u.

However, note that is homogeneous; because Note:

Solving this problem by direct calculation will be possible; but will involve lengthy calculations.

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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Homogeneous Functions and EulerтАЩs Theorem | Mathematics