I am confused about the definition of unique solution of ODE. I wonder which of following definitions ((1) or (2)) is correct?
Given an ODE $y'=f(x,y)$ and the initial condition $y(0) = 1, y'(0) = 2$ .
If $y=f(x)$ (defined on (-3,3) for example) satisfies the ODE and the initial condition then $y$ is said to be the unique solution on (-3,3) if
- For any function z(x) that satisfies the ODE and the initial condition on $(-3,3)$, we have $z(x) = y(x) \quad \forall x \in (-3,3)$ .
- For any function z(x) that satisfies the ODE and the initial condition on $(a,b) \subset (-3,3)$ and $ 0 \in (a,b)$, we have $z(x) = y(x) \quad \forall x \in (a,b)$ .
I mean if a solution $y_1$ if said to be unique on an interval $I$, is there any chance that there exists another solution $y_2$ defined on a subset $B$ of $(a,b)$ but does not coincide with $y_1$ on $B$? I think they are equivalent and tried to prove it by extending $y_2$ on $I$ but I am not sure if one can do that.