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) $ .