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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?

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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.

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  • $\begingroup$ thank you. will non homogenity of the equation make it possible for those functions to be solutions? $\endgroup$
    – jnyan
    Dec 4, 2017 at 5:41
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    $\begingroup$ Seems like both $x^3, x^2|x|$ satisfy the DE $xu'' - 2u' = 0$? $\endgroup$
    – BAYMAX
    Nov 11, 2019 at 22:18

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