If the Wronskian equals 0, is it always 0?

[TheCanadian]

If the Wronskian of a set of equations equals 0 over a particular interval in the functions' domain, is it possible for it be non-zero under another interval? Are there any particular proofs for or against this?

 

[Mark44]

If the Wronskian of a set of equations equals 0 over a particular interval in the functions' domain, is it possible for it be non-zero under another interval? Are there any particular proofs for or against this?

It took me a very long while to find a good example, but I found one in, of all places, "Advanced Engineering Mathematics", 3rd Ed., by Erwin Kreyszig.
Consider the three functions: $y_1 = x^3, y_2 = |x|^3, y_3 = 1$.
$W(y_1, y_2, y_3)$ is identically zero on one interval (implying that the three functions are linearly dependent on that interval), but $W(y_1, y_2, y_3)$ is different from zero on another interval (implying that the three functions are linearly independent on that other interval). I leave it to you to figure out what intervals we're talking about here.

 

[TheCanadian]

It took me a very long while to find a good example, but I found one in, of all places, "Advanced Engineering Mathematics", 3rd Ed., by Erwin Kreyszig.
Consider the three functions: $y_1 = x^3, y_2 = |x|^3, y_3 = 1$.
$W(y_1, y_2, y_3)$ is identically zero on one interval (implying that the three functions are linearly dependent on that interval), but $W(y_1, y_2, y_3)$ is different from zero on another interval (implying that the three functions are linearly independent on that other interval). I leave it to you to figure out what intervals we're talking about here.

Thank you! :D