Clarification on the Wronskian

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?

• 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 $I$. Nov 9 '16 at 7:58
• @CheeHan, Thanks, then what is the correct explanation for the second part? Nov 9 '16 at 8:09
• 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 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.
• 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. Nov 9 '16 at 15:58