the maximal interval of existence

Existence theorems are modest: they promise a solution only on some small interval around the start. But once you have a solution, you can usually push it further left and further right, extending it as far as it will honestly go. The maximal interval of existence is the largest open interval (a, b) containing the starting point x0 on which the solution lives — the full lifespan of that particular trajectory, beyond which it cannot be continued.

Think of it as following the trajectory until something stops you. There are only two ways a solution can fail to extend past some finite endpoint b: either the solution runs out of room in space — its value y(x) shoots off to infinity as x approaches b (a finite-time blow-up) — or it runs into the edge of the region where f is defined and well-behaved. As long as the solution stays inside a region where f is nice and the value stays bounded, the local existence theorem can be reapplied at the new frontier to push a little further; the maximal interval is exactly where that pushing finally becomes impossible. The endpoints a and b need not be symmetric, and either can be finite or infinite.

This is why it is misleading to ask 'what is the solution' without asking 'on what interval.' The same tidy-looking equation can yield a solution defined for all x or one that exists only on a short interval before exploding. The maximal interval is the honest home of the solution; reporting a formula without it can quietly overstate where the formula is valid. The width of this interval can also depend on the initial condition, so nudging the starting value can shorten or lengthen the life of the solution.