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Euler's Theorem on Homogeneous Functions

1. A function f(x,y) is homogeneous of degree n if f(λx, λy)=λ^n f(x,y) for all λ. 2. If a function f(x,y,z) is homogeneous of degree n, the Euler theorem states that xf/x + yf/y + zf/z = nf. 3. Several examples are provided of determining if functions are homogeneous and finding their degree, including proving homogeneous properties using the definition and Euler's theorem.

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Vinay Adari
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334 views11 pages

Euler's Theorem on Homogeneous Functions

1. A function f(x,y) is homogeneous of degree n if f(λx, λy)=λ^n f(x,y) for all λ. 2. If a function f(x,y,z) is homogeneous of degree n, the Euler theorem states that xf/x + yf/y + zf/z = nf. 3. Several examples are provided of determining if functions are homogeneous and finding their degree, including proving homogeneous properties using the definition and Euler's theorem.

Uploaded by

Vinay Adari
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Homogeneous function:

A function is said to be a homogeneous function of degree


if .
A function is said to be a homogeneous function of degree
if .

Note:
If is a homogeneous function of degree then we can
write as given below

or

Department of Mathematics

Department of Mathematics
If is a homogeneous function of degree in & , then
the theorem is
Proof:
Given that
Then w.k.t (1)

Now

(2)

and

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(3)
Adding the equations (2) & (3), we get

But from equation (1), we have

Note:
1. If , & , then

2. If is not a homogeneous function but is a homogeneous function


of degree , then &

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(1). Find to the following functions
(a). , (b). , (c).
(d). , (e).

Solution:
(a). Given function is
Now

is a homogeneous function of degree

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(c).
Solution:
Given function is

Now

is a homogeneous function of degree

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(e).
Solution:
Given function is
Now

is a homogeneous function of degree

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(2). If , then find


Solution:
Given function is

Now

is a homogeneous function of degree

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(3). If , then find
Solution:
Given function is

Now

is a homogeneous function of degree

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(4). If , then find


Solution:
Given function is

Let
Then
Put , ,

is a homogeneous function of degree


,

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(5). If , then prove that
Solution:
Given function is

Let
Then
Put ,

is a homogeneous function of degree

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, where

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(6). If , then show that
Solution:
Given function is

Let
Then
Put , ,

is a homogeneous function of degree

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, where

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(7). If , then find
Solution:
Given function is

Let
Then
Put ,

but is a homogeneous function of degree

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, where

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(8). If , then prove that
Solution:
Given function is

Let
Then
Put ,

is a homogeneous function of degree

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, where

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(9). If , then find
Solution:
Given function is

Let

Then
Put ,

is a homogeneous function of degree

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, where

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(10). If , then show that

(11). If , then prove that

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Dr. MADHUMOHANA RAJU A B, Associate Professor, Department Mathematics

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