Question

X3+x2y-xy2+y3/x2y+xy2

whetther homogeneous or not by dx/dy

Answer 1

Ans:-

$F(x,y)=\dfrac{x^3+x^2y-xy^2+{y^3}}{{x^2y}+xy^2}\\$

For finding equation homogeneous or not

Put $x=\lambda x$ and $y=\lambda y$

$F(\lambda x,\lambda y)=\dfrac{\lambda^3 x^3+\lambda^3 x^2 y-\lambda^3 xy^2+\lambda^3 y^3}{\lambda^3 x^2y+\lambda^3 xy^2}$

$\Rightarrow F(\lambda x,\lambda y)=\dfrac{x^3+x^2y-xy^2+{y^3}}{{x^2y}+xy^2}\\$

$\Rightarrow F(\lambda x,\lambda y)=F(x,y)$

Thus $F(x,y)$ is Homogeneous function of degree zero. Therefore, the given differential equation is homogeneous differential equation.