Wronskian
The Wronskian of is , the determinant of the fundamental matrix.
Definition 6.10 (Wronskian)
The Wronskian of the functions is defined as the determinant of the fundamental matrix:
Linear dependence test
If are linearly dependent, then for all : differentiating a total of times gives , so the columns of the Wronskian are linearly dependent at every .
It follows that if for some , then are linearly independent.
The converse fails: for all does not necessarily imply that are linearly dependent.
Example
For the two solutions , of ,
Since , the two solutions are linearly independent.
Related
Stated in
- Definition 6.10 (Wronskian)§6.3.3 Wronskian and Linear Dependence