Question

Consider the differential equation : $\frac{dy}{dx}=\frac{y^{3}}{2\left(xy^{2}-x^{2}\right)}$
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 $y^{2}e^{-\frac{y^2}{x}}=C.$

• A. Both statements are false
• B. Statement-1 is true and statement-2 is false
• C. Statement-1 is false and statement-2 is true
• D. Both statements are true

Answer 1

Answer: (D)

Given differential equation is
$\frac{dy}{dx}=\frac{y^{3}}{2\left(xy^{2}-x^{2}\right)}$
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, $\frac{dx}{dz}=\frac{x}{z}-\frac{x^{2}}{z^{2}}=\frac{x}{z}\left[1-\frac{x}{z}\right]\approx F\left(\frac{x}{z}\right)$
Hence, statement-1 is true.
Now, $y^{2}e^{-y^2/x}=C$ satisfies the given diff. equation
$\therefore $ It is the solution of given diff. equation. Thus, statement-2 is also true.