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
