Question #40729

The question tests the 'Modulus Trap' within a Linear Differential Equation.

Students must recognize that the modulus $|x-1|$ forces a piece-wise solution.

The critical trap lies in the 'Slope Sensitivity' and continuity: candidates often incorrectly apply the initial condition $y(0)=2$ directly to the $x \ge 1$ branch, bypassing the continuity requirement at $x=1$.

This leads to an incorrect constant of integration ($C=4$ instead of $2e$) and a false minimum of $2\ln 2 - 1$.

Proper execution requires solving the $x < 1$ branch first, using the limit at $x=1$ to find the continuity constant for the $x \ge 1$ branch, and then optimizing the resulting curve.