(d y)/(d x)=y^3/(2(x y^2-x^2)): Statement 1: The substitution z=y^2 transforms the above equation into a first order homogenous differential…

\section*{Consider the differential equation} $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y^3}{2\left(x y^2-x^2\right)}: $$ Statement $\displaystyle 1$ : The substitution $\displaystyle z=y^2$ transforms the above equation into a first order homogenous differential equation. Statement $\displaystyle 2$ : The solution of this differential equation is $\displaystyle y^2 \mathrm{e}^{-y^2 / x}=\mathrm{C}$.