Mathematics - Differential Equation Question with Solution | TestHub

Given differential equation is $$\frac{dy}{dx}=\frac{y^3}{2\left(x{y}^2-x^2\right)}$$ By substituting $z=y^2$, we get diff. eqn. as $$\frac{dz}{dx}=\frac{2z^2}{2\left(xz-x^2\right)}=\frac{z^2}{xz-x^2}$$ Now, $\frac{dx}{dz}=\frac{x}{z}-\frac{x^2}{z^2}=\frac{x}{z}\left[1-\frac{x}{z}\right]\approx \mathrm{F}\left(\frac{x}{z}\right)$ 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.