If $y_1(x)$ and $y_2(x)$ are two solutions of equation $y'' +P(x)y' +Q(x)y = 0$ on an interval $[a,b]$ and have a common zero in this interval, show that one is a constant multiple of the 1other.
Suppose the initial values of the solutions $y_1$ and $y_2$ are defined as follows : $y_1(t_o)=0,y_1'(t_o)=c_1$ and
$y_2(t_o)=0, y_2'(t_o)=c_2,$ then,
$y_1-y_2$ is also a solution to the given differential equation which satisfies :
$(y_1 -y_2 )(t_o)=0 , (y_1~ -y_2)~'(t_o) = c_1-c_2$.
Since, all of $y_1~,y_2,~y_1-y_2$ are solutions to he given differential equation, as per the uniqueness theorem, there exist unique curves which satisfy the above initial conditions.
Hence, I do not completely understand why $y_1$, $y_2$ must be a constant multiple of the other . I found an answer to this problem here. However, I do not understand the answer quite much.
Could someone please give an explanation to my confusion above.
Thank you for the help!