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How can I show that If the Wronskian of two solutions to a second-order ODE with $\textbf{no boundary conditions}$ is zero then they are linearly dependent?

Please do not mark this as duplicate as there is no answer to this particular question nowhere.

Lucas Pereiro
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  • That answer is wrong. Consider $f_1=x^2$ and $f_2=x|x|$. They have zero Wronskian, but are independent – Lucas Pereiro Jul 19 '19 at 15:47
  • It was the wrong link, sorry for that. See https://math.stackexchange.com/a/437750/669687. – Monadologie Jul 19 '19 at 16:06
  • There they only state what I am asking – Lucas Pereiro Jul 19 '19 at 16:07
  • $f_2=x|x|$ is not twice continuously differentiable and can thus not be a solution of a second order ODE. With any higher power you get that value and derivative at zero are zero, so still not a solution of a second order ODE whose domain includes $x=0$. – Lutz Lehmann Jul 21 '19 at 10:49

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