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.

| cite | improve this answer | |
  • $\begingroup$ thank you. will non homogenity of the equation make it possible for those functions to be solutions? $\endgroup$ – jnyan Dec 4 '17 at 5:41
  • $\begingroup$ Seems like both $x^3, x^2|x|$ satisfy the DE $xu'' - 2u' = 0$? $\endgroup$ – BAYMAX Nov 11 '19 at 22:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.