Given a set of fundamental solutions to an nth order linear homogeneous differential equation $y_1, y_2, y_3, ..., y_n$, can we prove that the Wronskian of this set is always non-zero for any value of x on the interval for which the solution set is defined?
I haven't been able to find a set of fundamental solutions whose Wronskian is zero for at least one value of x, and I'm not sure how to prove it. Conversely, I haven't been able to find a set of solutions that do not form a fundamental set, whose Wronskian is always non-zero.
As an example, if we look at the fundamental set $e^x, xe^x, x^2e^x$, it has a Wronskian of $2e^{3x}$. No value of x can make this Wronskian equal to zero. Conversely, the non-fundamental set $e^x, x^2e^x$ has a Wronskian $2xe^{2x}$, which can made zero using a substitution of x=0. I want to find out if this is true for all fundamental and non fundamental sets.
Any tips/proofs available?
This page mentions "Thereom 3" which seems like what I want, but without a proof. The Wronskian and the term "fundamental set of solutions"