The Wronskian of a set of functions is another function, which is zero over any interval where is linearly dependent. Just as a set of vectors is said to be linearly dependent when there exists a non-trivial linear relation between them, a set of functions is also said to be dependent over an interval when there exists a non-trivial linear relation between them, i.e.,
for some , not all zero, at any .
Therefore the Wronskian can be used to determine if functions are independent. This is useful in many situations. For example, if we wish to determine if two solutions of a second-order differential equation are independent, we may use the Wronskian.
Consider the functions , , and . Take the Wronskian:
Note that is always non-zero, so these functions are independent everywhere. Consider, however, and :
Here only when . Therefore and are independent except at .
Consider , , and :
Here is always zero, so these functions are always dependent. This is intuitively obvious, of course, since
Given linearly independant functions , we can use the Wronskian to construct a linear differential equation whose solution space is exactly the span of these functions. Namely, if satisfies the equation
then for some choice of .
Hence, the equation is which indeed has exactly polynomials of degree at most two as solutions.
|Date of creation||2013-03-22 12:22:59|
|Last modified on||2013-03-22 12:22:59|
|Last modified by||rspuzio (6075)|