Question

Observe the following statements :

$I$:If $z=xy\tan \left(\frac{y}{x}\right)$, then $x.\frac{\partial z}{\partial x}+y.\frac{\partial z}{\partial y}=z$

$II$: The degree of the homogeneous fucntion
$f(x,y)=\frac{1}{x^{2}y}+\frac{1}{x{y}^2}-\frac{\log x-\log y}{x^3+y^3}$ can not be defined.

Which of the above statements are correct?

• A. only $I$
• B. only $II$
• C. both $I$ and $II$
• D. Neither $I$ nor $II$

Answer 1

The correct option is D Neither $I$ nor $II$

$\frac{dz}{dx}=ytan\left(\frac{y}{x}\right)+xyse{c}^2\left(\frac{y}{x}\right)\left(\frac{-y}{x^2}\right)$
$\frac{dz}{dy}=xtan\left(\frac{y}{x}\right)+xyse{c}^2\left(\frac{y}{x}\right)\left(\frac{1}{x}\right)$
$x\frac{dz}{dy}+y\frac{dz}{dy}=xytan\left(\frac{y}{x}\right)-y^{2}se{c}^2\left(\frac{y}{x}\right)+xytan\left(xy\right)+y^{2}se{c}^2\left(\frac{y}{x}\right)$
$x\frac{dz}{dy}+y\frac{dz}{dy}=xytan\left(\frac{y}{x}\right)+xytan\left(xy\right)$
$=2z$
I is false
$f(x,y)=\frac{1}{x^{2}y}+\frac{1}{x{y}^2}-\frac{Log\frac{x}{y}}{x^3+y^3}$
it is homogenous function of power 3