Question #41351

This question tests the 'Cross Multiplication Trap' in homogeneous differential equations.

Students typically cross-multiply to get $dv/\sqrt{1-v^2} = dx/x$ and blindly integrate to find $y = x \sin(\ln x)$, leading to the incorrect answer $f(e^\pi) = 0$.

They fail to check the critical point $v=1$ (or $y=x$) where the denominator vanishes, which is a singular solution.

Furthermore, they ignore the non-negative range of the square root function ($x v' = \sqrt{1-v^2} \ge 0$), which dictates that $v(x)$ must be monotonically non-decreasing.

The true solution merges with the singular solution $y=x$ at $x=e^{\pi/2}$ and remains there, testing the student's ability to analyze the continuity and domain constraints of the tangent family.