The Wronskian of the functions \(y_1 = x^3\), \(y_2 = |x|^3\), and \(y_3 = 1\) demonstrates that it can equal zero on one interval while being non-zero on another. This indicates that the functions are linearly dependent in the first interval and linearly independent in the second. This example is sourced from "Advanced Engineering Mathematics", 3rd Ed., by Erwin Kreyszig, providing a definitive case for the behavior of the Wronskian across different intervals.
PREREQUISITESMathematics students, educators, and researchers interested in differential equations and linear algebra, particularly those studying the implications of the Wronskian in function analysis.
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.TheCanadian said: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 said: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.