Question #41475

This question tests the 'Zero-Factor Cancellation Trap' and the 'Sign-Consistency Principle' in the context of homogeneous differential equations.

When students substitute $y = vx$, they obtain $x v' = \sqrt{1-v^2}$.

Dividing by $\sqrt{1-v^2}$ implicitly assumes $v \neq 1$, which masks the singular solution $y=x$.

Furthermore, integrating blindly yields $y = x \sin(\ln x)$ for all $x$, but substituting this back into the original DE reveals a sign inconsistency for $x > e^{\pi/2}$ because $\sqrt{x^2-y^2}$ must be non-negative, forcing $\cos(\ln x) \ge 0$.

Recognizing that $v(x)$ reaches its maximum domain limit at $x = e^{\pi/2}$ and must smoothly transition to the singular branch $y=x$ is a hallmark of deep conceptual mastery required for JEE Advanced.