The promise was always local
Look back at the fine print of every theorem in this rung. Peano gave you a solution on some interval around x0. Picard-Lindelof, by adding the Lipschitz condition, upgraded that to a *unique* solution — but still only on some interval around x0. Read the Picard-Lindelof theorem honestly and you notice it never says "for all x". The guarantee it hands you is fundamentally local: a solution exists, and it is the only one, in a possibly tiny neighbourhood of the starting point.
That gap between the local guarantee and the global ambition is the whole subject of this final guide. You would love to know the solution lives for every x you might care about — but nothing so far promises it. The honest distinction is local versus global: a local solution exists near x0 by the theorems above; a global one would survive across the entire range you want. The leap from one to the other is not free, and seeing exactly why is the point of everything that follows.
Stitching the pieces: continuation
So how do you reach further than the first small interval? You re-apply the theorem. Suppose Picard-Lindelof gives you the solution up to some point x1, where it takes the value y1. Now treat (x1, y1) as a brand-new starting point and apply the theorem again: as long as f is still continuous and Lipschitz near (x1, y1), you get another little stretch of solution beyond x1. Glue it to the first piece. This stitching is the continuation of solutions, and it lets you keep extending step by step.
Push this process as far as it will go and you arrive at the maximal interval of existence — the single largest open interval, say (a, b) around x0, on which the unique solution lives. By construction it cannot be extended any further: it is the solution's full natural lifespan, the answer to "how long does it last?" The maximal interval is uniquely determined by the equation and the initial condition together; it is not something you choose, it is something the problem decides for you.
Finite-time blow-up: the dramatic exit
The second way out — the value racing to infinity — has a vivid name: finite-time blow-up. The cleanest example is y' = y^2 with y(0) = 1. The right-hand side f(x, y) = y^2 is smooth and Lipschitz on any bounded region, so Picard-Lindelof happily gives a unique local solution. Separate variables and you find y(x) = 1/(1 - x). Watch what happens: as x climbs toward 1, the denominator shrinks to zero and y shoots to infinity. The solution simply ceases to exist at x = 1, even though the equation itself looks perfectly tame.
y' = y^2, y(0) = 1 separate: dy/y^2 = dx -> -1/y = x + C fit y(0)=1: C = -1 -> y(x) = 1 / (1 - x) x -> 1^- => y -> +infinity (BLOW-UP at x = 1) maximal interval of existence: (-infinity, 1)
Two lessons land hard here. First, the local theorems told the whole truth: they only ever promised a solution near x = 0, and indeed the solution exists on (-infinity, 1) and not an inch beyond. Second — and this is the misconception to kill — a smooth, Lipschitz, beautifully unique right-hand side does not guarantee a global solution. Finite-time blow-up is the everyday reason that a perfectly well-posed initial value problem can still have a solution that lasts only a finite while. Niceness of f controls existence and uniqueness; it does not, by itself, control lifespan.
What forces a solution to last forever
If niceness of f is not enough, what is? The cleanest sufficient condition is a growth cap. If f(x, y) grows no faster than linearly in y — concretely, |f(x, y)| stays below something like A + B|y| — then the solution simply cannot reach infinity in finite time, and the maximal interval is the whole line. The reason y^2 blows up is exactly that it grows *quadratically*: feedback that strong outruns any finite clock. Slow the feedback to linear and the explosion is forbidden.
The tool that makes this rigorous is Gronwall's inequality. Roughly, it says that if a quantity's growth rate is bounded by a multiple of the quantity itself, then the quantity can grow at most exponentially — and exponential growth, however fast, still takes infinitely long to reach infinity. So a linear growth bound on f feeds into Gronwall's inequality and out comes an exponential ceiling on |y|, which keeps the solution finite at every finite x. This is also why every linear equation y' = p(x) y + q(x) with continuous coefficients has solutions on the whole interval where the coefficients live: linearity *is* a linear growth bound.
Honesty check: a growth bound is *sufficient* for global existence, not necessary. The logistic equation y' = r y (1 - y/K) has a quadratic right-hand side too, yet its solutions never blow up — the negative feedback past the carrying capacity bends every trajectory back toward K. So the real question is never just "how big is f" but "does the feedback push the solution outward fast enough to escape". Quadratic-and-outward explodes; quadratic-but-self-limiting survives.
Why the interval question closes the rung
Step back and see the three promises hidden inside the innocent phrase "solve the IVP", now fully separated. Existence: Peano, from continuity alone. Uniqueness: Picard-Lindelof, once you add the Lipschitz condition. And lifespan: the maximal interval, possibly cut short by blow-up. Each promise needed its own hypothesis and its own argument, and no single one implies the others. That clean dissection is the real prize of this whole rung.
There is one more promise worth naming, because the next rungs lean on it: that the solution depends *continuously* on the data. Nudge the initial condition a little and the solution shifts only a little, at least on a finite interval — this is continuous dependence on initial conditions, and it too falls out of Gronwall's inequality. Existence plus uniqueness plus continuous dependence together is exactly what it means for the problem to be well-posed. Well-posedness is the formal license that says: yes, go ahead and solve this on a computer and trust what you get.
And that is the doorway out of this rung. Now that you trust a unique solution exists on a definite interval and bends predictably with the data, you are licensed to chase it. The numerical methods later in this ladder approximate that very solution step by step; the qualitative phase-plane reasoning describes its shape without a formula. Both rest on the certificate you just earned — that there is exactly one solution there to be found, and you know, honestly, how long it lasts.