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I understand that if $~y_1,y_2,\cdots,y_n~$ are solutions of a normalized homogenous linear differential equation in $~I~$, then the wronskian of the solutions is always $~0~$ or never $~0~$ for every $~x~$ in $~I~$.

My question is: If I have that the wronskian of two functions is always zero or never zero in $~I~$, can I say that those functions are solutions of some normalized homogenous linear differential equation in $~I~$?

Thanks

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  • $\begingroup$ Can you be more clear: what do you mean by "normalized homogenous L D E ? Maybe an example? $\endgroup$ – orangeskid Oct 22 '17 at 0:31
  • $\begingroup$ @orangeskid The linear differential equation of n orders that is equaled to 0 and the coefficient of the nth derivative is different of 0. Something like y’’+2y’-3y=0 $\endgroup$ – Danilo Ch Oct 22 '17 at 0:48
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If the Wronskian of $y_1$, $y_2$ is not $0$ on $I$ then $y_1$, $y_2$ are solutions of a linear equation of form $$y''(t) + a_1(t) y'(t) + a_2(t) y(t)= 0$$

That is not very hard to show. Note that you cannot guarantee that $a_1$, $a_2$ are constants.

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