Consider dydx=y^32(xy^2-x^2).

Question

Consider $\dfrac{dy}{dx}=\dfrac{y^3}{2(xy^2-x^2)}$. Statement-1: $z=y^2$ transforms this into a first-order homogeneous equation. Statement-2: the solution is $y^2e^{-y^2/x}=C$.

Accepted Answer

Substituting $z=y^2$ indeed makes the equation homogeneous in $x,z$, and direct verification shows $y^2e^{-y^2/x}=C$ satisfies the ODE --- so both statements are true.

Solutions · 1