Non-singular solution of the differential equation $x\frac{dy}{dx} = x - \left(\frac{dy}{dx}\right)^2$ is :
Step-by-Step Solution
Key Concept: Recognizing Clairaut's form allows immediate application of the standard solution method.
Setting $\frac{dy}{dx} = p$, the equation becomes $y = xp + p^2$, which is Clairaut's equation. The general solution is obtained by replacing $p$ with arbitrary constant $c$: $y = cx + c^2$.
Correct Answer: 2