We saw yesterday that when we're solving second order linear homogeneous differential equations with constant coefficients, we will often end up with two solutions. We can use these solutions to find a two-parameter family of solutions by taking a linear combination of them. What makes these two solutions special? The answer lies with the concept of linear independence.
Definition: A set of functions is linearly dependent on an interval if there are constants that are not all zero so that
for all The set of functions is linearly independent if it is not linearly dependent.
Example: The set is linearly dependent on the interval since picking we have
Example: The set are linearly independent on
Suppose the set was linearly dependent, then we could find constants so that
for all Then at we have
So Similarly, at we would have
and therefore Using this at gives us
which implies that Since all the constants are zero, and the set is linearly independent.
Looking at this example in retrospect, choosing three values of got us three equations for the three unknown constants . We can rewrite these three equations as a linear system
By the invertible matrix theorem from linear algebra, this system has a nonzero solution if and only if the determinant of the matrix is not zero. The determinant of this matrix is , and therefore the only solution is
This previous solution required us to make some choices about which values of we wanted to use. However, if the functions are differentiable on the whole interval then we can get our three equations by taking derivatives of the functions instead. In this case, we have
which can be written as the linear system
This matrix is upper triangular, so the determinant is just the product of the diagonal entries. Since none of those entries are 0, the determinant is non zero and therefore the only solution is We can generalize this idea to an arbitrary set of functions:
Definition: Let be a set of functions that are times differentiable. The Wronskiian for the set is
If for at least one choice of , then is linearly independent on
Example: Is the set linearly independent on ?
Solution:
Since the set is linearly independent.
Practice Problems:
- Show that is linearly dependent on
- Show that is linearly independent on Sketch a graph of the two functions to illustrate why the interval is important.
- Determine whether or not is linearly independent.
- Determine whether or not is linearly independent.