Wronskian of $x^3$ and $x^2 |x|$

The theorem says if the functions are linearly dependent, then the Wronskian is $0$. It does not state that if the Wronskian is $0$, the functions are linearly dependent. These functions are an example that shows this. Their Wronskian is $0$, and they would be linearly dependent if you just looked at the interval $(-\infty, 0)$ or $(0, \infty)$, but they are not linearly dependent on the whole real line because neither is a constant multiple of the other.

The point (which will probably be made later) is that solutions of homogeneous second-order linear differential equations are rather special: if they are linearly dependent in some interval, this same linear dependence holds over the whole interval where the differential equation is defined; moreover, the value of the Wronskian at one point tells the whole story.