Question #40396

This question tests the intersection of complex binomial expansions and the topological domain of homogeneous differential equations.

The primary trap lies in the substitution $v = y/x$.

Students typically integrate the separated variables to find the algebraic curve $16x^3 = v(3-v^2)^4$ and incorrectly assume $v$ can take any positive value since the equation yields a real $x$.

However, the periodic nature of the allied angle coefficients masks the fact that $\frac{Q}{P} = \tan(3 \arctan v)$, which introduces a singularity at $v = \sqrt{3}$ where $x \to 0$.

Because the curve is restricted to the first quadrant ($x > 0$), it cannot cross this singular boundary.

Failing to explicitly constrain the periodicity of the solution branch leads students to erroneously accept values like $7/4$ or $2$, missing that the valid domain for this specific branch is strictly $v \in (0, \sqrt{3})$.