the numerical initial value problem

Most differential equations have no formula for their solution. You can write down the rule y' = f(t, y) and the starting point y(t0) = y0, and yet there may be no combination of familiar functions — no exponentials, sines, logarithms — that solves it. So what do you do? You do what a hiker with a compass but no map does: you start where you are, look at which way the ground tilts, take a small step in that direction, then look again. Numerical methods turn a differential equation into exactly this kind of step-by-step walk.

Precisely, the numerical initial value problem is this setup: given f(t, y), a starting time t0, and a starting value y0, produce approximate values y1, y2, y3, ... of the true solution at later times t1, t2, t3, ..., usually spaced a step h apart so t_(n+1) = t_n + h. We never get the whole function y(t); we get a table of numbers, one per step, that traces it out. The differential equation tells us the slope f(t_n, y_n) at any point we are standing on, and every method is some clever rule for using that slope information to guess the next value y_(n+1) from the current one y_n.

This is how almost all real differential equations are actually solved — in weather models, spacecraft trajectories, drug-dosing simulations, circuit design. The analytic methods of the rest of this subject solve a small, special list of equations exactly; numerical methods solve everything else, approximately but well. The cost is honesty: a numerical solution is never exact. Two separate errors creep in — the error of approximating the step (truncation) and the error of finite-precision arithmetic (round-off) — and a good practitioner always knows roughly how big they are.