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?
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.