Question #41124

The trap tests the student's ability to handle implicit to explicit conversions involving variable transformations ($u = e^{f(x)}$).

The differential equation factors into two branches.

The 'Lethal Collision' occurs when students blindly integrate both branches and fail to check the global domain constraints.

The negative branch yields a function that eventually becomes negative, making the logarithmic back-substitution undefined and thus failing the requirement that $f$ is defined on all of $\mathbb{R}$.

Furthermore, the twice-differentiable condition strictly prohibits piecewise branch switching at the critical point $x=0$ (as $u''(0)$ differs between branches).

Unwary students will either assume both branches are valid and guess $(-\infty, \infty)$, or fail to properly minimize the correct branch, falling into a piecewise logic error.