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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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