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
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.