Why is it that solutions $x_1 \text{ and } x_2 \text{ to a differential equation are linearly dependent } \iff W(x_1,x_2)(t_0)=0$

Question

Can someone explain from where the aforementioned fact is derived? By the way $W = \text { Wronskian determinant}.$

Answer 1

For the non-trivial direction, suppose $x_1$ and $x_2$ are solutions to $$x''+bx'+cx=0,$$define $$W=x_1x_2'-x_1'x_2,$$and assume $$W(t_0)=0.$$Define $$y(t)=x_2'(t_0)x_1(t)-x_1'(t_0)x_2(t).$$Of course $y$ is a solution to the original DE, being a linear combination of $x_1$ and $x_2$. It's clear that $y'(t_0)=0$, and $W(t_0)=0$ shows that $y(t_0)=0$. So uniqueness shows that $$y(t)=0,$$which says that $x_1$ and $x_2$ are dependent.