So as I understand, the Wronskian determinant can be used to show linear independence. Why is that? Also, how does this fit into understanding variation of parameters for solving a differential equation?
[Math] the purpose of Wronskian (linear independence/variation of parameters)
linear algebraordinary differential equations
Best Answer
The reason that the Wronskian can be used to determine linear dependence is because if a group of functions are linearly dependent then so are their $n$th derivatives. Hence, if we take the determinant of a matrix of column vectors which each consist of the $n$th derivatives of a given function in each of their rows, the matrix will consist of column vectors which are linearly dependent (and therefore the determinant will be zero). It is NOT true that the Wronskian not being identically zero implies linear independence, however. The Wronskian pops up all over the place in linear differential equations. Abel's formula is one of the most useful tricks in an all of linear ODEs. It relies on the fact that there are a finite number of functions which span the kernel of a linear differential operator and that the Wronskian can be derived purely from the differential operator. We can therefore use the Wronskian to solve for unknown kernel basis functions. The general solution for second order linear differential equations (Green's Function, which is the general form solution of the variation of parameters) involves the Wronskian because the Wronskian "normalizes" various interactions much in the same way that the determinant of a traditional matrix is used to "normalize" an inverse matrix. I hope that helps and let me know if you have any more questions!