Question

Check whether the differential equation $(xy)dx - ({x^3} - {y^3})dy = 0$ is homogeneous or not.

Answer 1

Hint : Firstly, we will convert the given equation into the form of $\dfrac{{dy}}{{dx}}$ . Then we will assume $\dfrac{{dy}}{{dx}} = F(x,y)$ . Further we will find ${\lambda}F(x,y)$ .Thereafter we will check if it is homogeneous or not.

Complete step-by-step answer :
The given differential equation is
$xydx - ({x^3} + {y^3})dy = 0$
$xy\,dx = 0 + ({x^3} + {y^3})dy$
$\dfrac{{dy}}{{dx}} = \dfrac{{xy}}{{{x^3} + {y^3}}}$
$\Rightarrow \dfrac{{xy}}{{{x^3} + {y^3}}} = \dfrac{{dy}}{{dx}}$
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{xy}}{{{x^3} + {y^3}}}$
Let $F(x,y) = \dfrac{{dy}}{{dx}} = \dfrac{{xy}}{{{x^3} + {y^3}}}$
Now, I will check, it is homogeneous or not. We will put $\lambda$ to $xandy$ in the above $F\left( {x,y} \right)$ ,we will get $$
$F(\lambda x,\lambda y) = \dfrac{{{\lambda ^2}xy}}{{{\lambda ^3}{x^3} + {\lambda ^3}{y^3}}}$
Taking common ${\lambda ^3}$ in denominator, we have
$F\left( {\lambda x,\lambda y} \right) = \dfrac{{{\lambda ^2}xy}}{{{\lambda ^3}\left( {{x^3} + {y^3}} \right)}}$
$F(\lambda x,\lambda y) = \dfrac{{xy}}{{\lambda ({x^3} + {y^3})}}$
$\ne {\lambda}F(x,y)$
$\therefore$ The given equation is not homogeneous

Note : Students must know that if the given equation is a homogeneous then $f(x,y) = {\lambda}F(x,y)$ and if the given equation is not homogeneous then $F(x,y) \ne {\lambda}F(x,y)$ .