Wronskian

The Wronskian generalizes the technique for showing a set is linearly independent by showing they form a singular matrix matrix

to vector spaces of differentiable functions.¹ A Wonskrian of zero need not imply linear dependence…

Other notes

[!] The Wronskian is identically nonzero iff the solutions are linearly independent. A Wronskian of zero at a specific point indicates there is no solution at that point.
[i] As a consequence of this, it is possible to determine whether a specific function is even eligible to be one of the basis solutions: if both it and its derivative are zero at a specific point , the Wronskian will be zero at that point (irregardless of the other basis solution*) and hence it is not a solution at this point.