Question

Which of the following functions are not homogeneous?

• A. $x+y\cos \frac{y}{x}$
• B. $\frac{xy}{x+y^2}$
• C. $\frac{x-y\cos x}{y\sin x+y}$
• D. $\frac{x}{y}\ln \left(\frac{y}{x}\right)+\frac{y}{x}\ln \left(\frac{x}{y}\right)$

Answer 1

Answer: (B), (C)

The correct options are
B $\frac{xy}{x+y^2}$
C $\frac{x-y\cos x}{y\sin x+y}$

Consider $f\left(x\right)$ to be homogeneous.
Then it should satisfy the condition, $f\left(vx\right)=vf\left(x\right)$ or in other words, replacing $x$ by $\left(vx\right)$ and $y$ by $\left(vy\right)$ should give us $v$ times the same initial expression, where $v$ is an arbitrary constant.
For example consider option $A$
Replacing $x$ by $\left(vx\right)$ and $y$ by $\left(vy\right)$
$vx+vy\cos \left(\frac{vy}{vx}\right)$
$=v(x+y\cos \left(\frac{y}{x}\right))$
The degree of all the terms is same.
However these conditions are not met by B and C.
Hence Options B and C are non-homogenous.