I have this doubt since long time. I came across this statement in a book that two solutions to a linear second order homogenous equation are linearly dependent on an interval if and only if their wronskian is zero on that interval.
$x^3$ and $x^2|x|$ are linearly independent on $[-1,1]$ but they have $0$ wronskian. So would that mean they cannot be fundamental solutions to same second order linear differential equation defined on $[-1,1]$? is there any way to prove they cannot be solutions? but if they are solutions, would'nt that contradict the statement?