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Homogeneous Function Definition, Examples

What is Homogeneous Function Definition:

A function f defined by

u=f(x,y,z,...)

of any number of variables are said to be homogeneous of degree n in these variables if multiplication of these variables by any number t\neq0 result in the multiplication of the function by t^n \;\;,i.e,.

f(tx,ty,tz,...)= t^n f(x,y,z,...)\;\;\;\;\;\;\;\;\;(i)

provide that (tx,ty,tz,...) is in the domain of f.

Taking t=\frac 1x ,\;\; x\neq0 the equation(i) becomes

f\left(1,\;\frac yx,\frac zx,…\right)\;=\frac1{x^n}f\left(x,y,z,…\right).

or\;\;\;f\left(x,y,z,…\right)=x^nf\left(1,\;\frac yx,\frac zx,…\right).

\Rightarrow\;f\left(x,y,z,…\right)=x^ng\left(\frac yx,\frac zx,…\right).

which is another criterion for a function to be in a homogeneous function. so after the homogeneous function definition now come to the examples

Homogeneous Function Example: 1

consider the function f defined by

f\left(x,y\right)=\frac xy+\frac34\frac yx+\cos\sqrt{\frac yx}+\ln\left(x\right)-\ln\left(y\right).

f\left(tx,ty\right)=\frac{tx}{ty}+\frac34\frac{ty}{tx}+\cos\sqrt{\frac{ty}{tx}}+\ln\left(tx\right)-\ln\left(ty\right).

f\left(tx,ty\right)=\frac{tx}{ty}+\frac34\frac{ty}{tx}+\cos\sqrt{\frac{ty}{tx}}+\ln\;t+\ln\;x-\ln t\;-\ln\;y.

\Rightarrow f\left(tx,ty\right)=\frac xy+\frac34\frac yx+\cos\sqrt{\frac yx}+\ln\;x\;-\ln\;y.

\Rightarrow f\left(tx,ty\right)=f\left(x,y\right)=t^0f\left(x,y\right).

Thus f is homogenous function of degree 0.

Homogeneous Function Example: 2

Let

f(x,y)=\frac{\sqrt y+\sqrt x}{y+x}.

Here

f(x,y)=\frac{\sqrt y+\sqrt x}{y+x}=\frac{\sqrt x\left[1+\sqrt{\displaystyle\frac yx}\;\right]}{x\left[1+{\displaystyle\frac yx}\right]}.

f(x,y)=x^\frac{-1}2\frac{\left[1+\sqrt{\displaystyle\frac yx}\;\right]}{\left[1+{\displaystyle\frac yx}\right]}.

Thus f(x,y) is a homogeneous function of degree \frac{-1}2.

Another Method:

Let

f(x,y)=\frac{\sqrt y+\sqrt x}{y+x}.

Then

f(tx,ty)=\frac{\sqrt{ty}+\sqrt{tx}}{ty+tx}.

f(tx,ty)=\frac{t^{\displaystyle\frac12}\left(\sqrt y+\sqrt x\right)}{t\;\left(y+x\right)}

f(tx,ty)=\frac{t^{\displaystyle\frac{-1}2}\left(\sqrt y+\sqrt x\right)}{\;\left(y+x\right)}.

f(tx,ty)=t^\frac{-1}2f(x,y).

Thus f(x,y) is a homogeneous function of degree \frac{-1}2.

also check homogeneous equation you can also see this topic from here

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