Just to make it explicit which theorem i am considering here, let me first state the Existence and Uniqueness theorem for a second order linear differential equation.
Theorem: Consider an equation of the form
$$a_0 y'' + a_1 y' + a_2 y = r(x),$$
where $a_0(x)$, $a_1(x)$, $a_2(x)$ and $r(x)$ are continuous functions on an interval $(a,b)$ and $a_0(x) \ne 0$ for each $x \in (a,b)$. Let $c_1$ and $c_2$ be arbitrary real numbers and $x_0 \in (a,b)$. Then there exists a unique solution $y(x)$ defined over $(a,b)$ for the above equation satisfying $y(x_0) = c_1$ and $y'(x_0) = c_2$.
Now the author also mentions below corollary to the above mentioned theorem.
Corollary: If $y(x)$ be a solution to
$$a_0 y'' + a_1 y' + a_2 y = 0$$
(note $r(x) = 0$ here) satisfying $y(x_0) = 0$ and $y'(x_0) = 0$ for some $x_0 \in (a,b)$, then $y(x)$ is identically zero on $(a,b)$.
I am not able to convince myself as to why this corollary is valid. Need help with an informal proof to understand why this corollary holds good.
Thanks in advance.