## Wronskian

Given a set of

*n*functions

*f*, ...,

_{1}*f*, the Wronskian

_{n}*W(f*is given by:

_{1}, ..., f_{n})The Wronskian can be used to determine whether a set of differentiable functions is linearly independent on a given interval:

- If the Wronskian is non-zero at some point in an interval,
then the associated functions are
*linearly independent*on the interval.

The Wronskian is particularly beneficial for determining linear independence of solutions to differential equations. For example, if we wish to verify two solutions of a second-order differential equation are independent, we may use the Wronskian, which requires computation of a 2 x 2 determinant.

Note:
If the Wronskian is uniformly zero over the interval, the functions may
or may not be linearly independent. A common misconception is that *W* = 0
everywhere implies linear dependence.

Example:

Find the Wronskian for the functions sin x and cos x. Using value of the Wronskian, determine whether the functions are linearly independent.

Solution: