I am getting a few "contradicting" conclusions from the Wronskian and I just wanted to clarify, but assume $y_1, y_2$ are two solutions to a second order differential equation that is homogeneous.

According to my textbook:

If the Wronskian, $W\neq 0$ at some point $t_0$ on the interval $I$, then $y_1, y_2$ are linearly independent and form the fundamental set of solutions

But the solution to the question "Show that if $y_1, y_2$ are solutions for some second order differential equation, but they have a maxima/minima at the same point in $I$, then they cannot form the fundamental set of solutions" seems to contradict this. Their explanation is that:

Pick $t_0$, then the $W(y_1, y_2)(t_0) = 0$ at $t_0$, so they cannot be linearly independent, so they don't form the set of fundamental solutions

But this doesn't prove that $W=0$ across the entire interval $I$, so I'm confused. Could someone please enlighten me?

  • $\begingroup$ You were right. The first theorem that you stated about Wronskian is correct. The explanation for the question is wrong however: there exists linearly independent functions that have zero Wronskian over $I$. $\endgroup$
    – Chee Han
    Nov 9 '16 at 7:58
  • $\begingroup$ @CheeHan, Thanks, then what is the correct explanation for the second part? $\endgroup$
    – q.Then
    Nov 9 '16 at 8:09
  • $\begingroup$ I probably should have mentioned this, it follows from Abel's identity that Wronskian for a second order homogeneous linear ODE is either zero or never zero. But I don't have a good answer to the second part, been trying to come up with an example but failed :O $\endgroup$
    – Chee Han
    Nov 9 '16 at 8:32

At the maxima position $t_0$, the first derivatives $y_k(t_0)$, $k=1,2$ are zero. Assuming that none of the solutions is the trivial solution, you get $y_2(t_0)=cy_1(t_0)$ for some $c\ne 0$. Which means that the initial value vectors $(y_k(t_0),y_k'(t_0))$, $k=1,2$ for $t=t_0$ are multiples of each other which is then true for the full solution, $y_2(t)=cy_1(t)$ for all $t$ in the domain.

  • $\begingroup$ It could be worthwhile to throw in a reference to Abel's identity, as @CheeHan already commented on, showing that the Wronkskian (for a second order homogeneous linear ODE) is zero at some "time" $t$ if and only if zero identically. $\endgroup$
    – hardmath
    Nov 9 '16 at 15:58

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