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Which of the following is not a homogeneous function?

Option 1)
$\delta \left ( x,y \right )= x^{3}y+xy^{3}$

Option 2)
$\delta \left ( x,y \right )= \sqrt{\frac{x^{3}}{y}}+\sqrt{\frac{y^{3}}{x}}$

Option 3)
$\delta \left ( x,y \right )=x^{2}y-yx^{3}$

Option 4)
$\delta \left ( x,y \right )=x^{2}y+yx^{2}$

Answers (1)

best_answer

As we learnt

Homogeneous Differential Equation -

A function

$f\left ( x,y \right )$

is called homogeneous function of degree n, if

$f\left (dx,dy \right )=d^{n}\left ( x,y \right )$

- wherein

eg:

$f\left ( x,y \right )=x^{2}y^{2}-xy^{3}$

$(A)\rightarrow \delta \left ( dx,dy \right )= d^{3}x^{3}dy+dx.d^{3}y^{3}= d^{4}\left (x^{3}y+xy^{3} \right )$

$\Rightarrow \delta \left ( dx,dy \right )= d^{4}\delta \left ( x,y \right )$

$(B)\rightarrow \delta \left ( dx,dy \right )= \sqrt{\frac{\left ( dx \right )^{3}}{dy}}+\sqrt{\frac{\left ( dy \right )^{3}}{dx}}= d\sqrt{\frac{x^{3}}{y}}+d\sqrt{\frac{y^{3}}{x}}= d\delta \left ( x,y \right )$

$(C)\rightarrow \delta \left ( dx,dy \right )= d^{2}x^{2}dy-dy.d^{3}x^{3}= d^{3}x^{2}y-d^{4}yx^{3}$

$\Rightarrow \delta \left ( dx,dy \right )$ cant be expressed as $d^{4}\delta \left ( x,y \right )$ here.

$(D)\rightarrow \delta \left ( dx,dy \right )= d^{2}x^{2}dy+dy d^{2}x^{2}= d^{3}x^{2}y+d^{3}yx^{2}= d^{3}\left ( x^{2}y+ yx^{2}\right )$

$\Rightarrow \delta \left ( dx,dy \right )= d^{3} \delta \left ( x,y \right )$

$\therefore (A),(B),(D)$ are homogeneous but (C) is not.

Option 1)

$\delta \left ( x,y \right )= x^{3}y+xy^{3}$

Option 2)

$\delta \left ( x,y \right )= \sqrt{\frac{x^{3}}{y}}+\sqrt{\frac{y^{3}}{x}}$

Option 3)

$\delta \left ( x,y \right )=x^{2}y-yx^{3}$

Option 4)

$\delta \left ( x,y \right )=x^{2}y+yx^{2}$

Posted by

Himanshu

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