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.
The Wronskian is a mathematical concept used in differential equations to determine whether a set of functions is linearly independent or dependent.
If the Wronskian equals 0, it means that the set of functions is linearly dependent, which means that one function can be expressed as a linear combination of the other functions in the set.
No, the Wronskian is not always 0. It can only be 0 if the set of functions is linearly dependent.
Yes, the Wronskian can be negative, positive, or 0. It is dependent on the specific set of functions being evaluated.
If the Wronskian is 0, it indicates that the set of functions is not a fundamental set of solutions for a given differential equation. This means that the solutions to the differential equation may not be unique and further analysis is needed.