# Wronskian for functions

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

• Can you be more clear: what do you mean by "normalized homogenous L D E ? Maybe an example? – orangeskid Oct 22 '17 at 0:31
• @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 – Danilo Ch Oct 22 '17 at 0:48

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.