the Picard-Lindelof theorem
/ pee-KAR LIN-de-loff /
This is the workhorse existence-and-uniqueness theorem for first-order initial value problems, and the one most textbooks lean on. It says: if the right-hand side f(x, y) is continuous in a box around the starting point (x0, y0) and satisfies a Lipschitz condition in y there, then the problem y' = f(x, y), y(x0) = y0 has exactly one solution on some interval around x0. Continuity buys existence; the Lipschitz condition upgrades it to a single, unambiguous solution. Both halves are delivered together, locally.
Here is how the proof actually runs, because the method is as valuable as the conclusion. You rewrite the differential equation as an equivalent integral equation, y(x) = y0 + integral from x0 to x of f(t, y(t)) dt, so that solving the ODE becomes finding a function that equals this expression — a fixed point of the map that sends a candidate function to the right-hand side. On a short enough interval the Lipschitz constant makes that map a contraction (it shrinks distances between functions), and the contraction-mapping principle guarantees exactly one fixed point. You reach it by Picard iteration: start with a guess, plug it into the integral to get a better one, and repeat; the successive approximations converge to the unique solution.
Two honest caveats keep this from being oversold. First, it is local: the theorem only promises a solution on some possibly small interval, not for all x — the solution may still blow up later. Second, the Lipschitz hypothesis is essential, not decorative; drop it and uniqueness can genuinely fail, as y' = y^(2/3) through the origin shows. So Picard-Lindelof is best understood as: enough smoothness to forbid branching, plus a constructive recipe to build the one true solution.
For y' = x + y, y(0) = 1, the right-hand side is smooth and Lipschitz in y (slope in y is just 1 everywhere), so Picard-Lindelof gives a unique solution near 0 — and Picard iteration starting from y0(x) = 1 generates the partial sums of that solution one term at a time.
Continuity plus a Lipschitz bound yields one local solution, and the proof hands you a way to build it.
The guarantee is local. A perfectly nice, smooth right-hand side can still produce a solution that exists only on a short interval before blowing up — uniqueness near the start says nothing about how long the solution lives.