Solution of the differential equation $x^2\left(y - x\frac{dy}{dx}\right) = y\left(\frac{dy}{dx}\right)^2$ which does not contain singular solution is :
Step-by-Step Solution
Key Concept: Recognize that substituting $p = \frac{dy}{dx}$ and rewriting as $x^2(y - xp) = yp^2$ transforms the equation into Clairaut's form $v = uc + c^2$ when using substitutions $u = x^2, v = y^2$. The general solution $y^2 = cx^2 + c^2$ emerges directly without introducing extraneous singular solutions.
Substitute $x^2 = u$ and $y^2 = v$ to transform the differential equation. This gives $\frac{dy}{dx} = \frac{x}{y}\left(\frac{dv}{du}\right)$. The resulting equation becomes Clairaut's equation in variables $u$ and $v$: $v = uc + c^2$ or $y^2 = cx^2 + c^2$, which is the general solution.
Correct Answer: 3