Refer to following question- Determine the *interval* in which the solution is defined?
In context of the solution by Florain, I have some doubts.
By uniqueness and existence theorem, we can say that for $y'=f(x,y)$, an initial value problem on the region where $f(x,y)$ is continuous, we will have a unique solution. But what is asked in the question is that solution exist, uniqueness is not required. Does this means that a solution is not valid only because it is not unique? Consider the interval $(-1,2)\cup (4,\infty)$ for the same solution. It seems to me that it satisfies the equation at all points in the interval. Is this solution not valid in this interval only because it is not unique?
Please correct me if any of my statement is incorrect.