# can linearly independent functions with zero wronskian be solutions to homogenous differential equation?

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?

This example is dealt with in Birkhoff and Rota's Ordinary Differential Equations. They observe that these two functions satisfy both $xu'=3u$ and $3xu''-2u'=0$.
They prove that this phenomenon doesn't occur for equations $u''+p(x)u'+q(x)=0$, $p$, $q$ continuous.
• Seems like both $x^3, x^2|x|$ satisfy the DE $xu'' - 2u' = 0$? – BAYMAX Nov 11 '19 at 22:18