Consider the differential equation :
$$
\frac{d y}{d x}=\frac{y^3}{2\left(x y^2-x^2\right)}
$$
Statement-1: The substitution $z=y^2$ transforms the above equation into a first order homogenous differential equation.
Statement-2: The solution of this differential equation is $y^2 e^{-y^2} / x=C$.
Both statements are false.
Statement-1 is true and statement- 2 is false.
Statement-1 is false and statement-2 is true.
Both statements are true.
Solution
Given differential equation is
$
\frac{d y}{d x}=\frac{y^3}{2\left(x y^2-x^2\right)}
$
By substituting $z=y^2$, we get diff. eqn. as
$
\frac{d z}{d x}=\frac{2 z^2}{2\left(x z-x^2\right)}=\frac{z^2}{x z-x^2}
$
Now, $\frac{d x}{d z}=\frac{x}{z}-\frac{x^2}{z^2}=\frac{x}{z}\left[1-\frac{x}{z}\right] \approx \mathrm{F}\left(\frac{x}{z}\right)$
Hence, statement-1 is true.
Now, $y^2 e^{-y^2 / x}=\mathrm{C}$ satisfies the given diff. equation
$\therefore$ It is the solution of given diff. equation. Thus, statement $-2$ is also true