the initial-value problem
Picture a ball rolling along a track where, at every point, the slope of the track tells you which way and how fast the ball is heading. If someone tells you exactly where the ball starts, you can imagine letting it go and watching the whole path unfold. An initial-value problem is precisely this: a rule for the rate of change at every moment, plus a single known starting state, and the question of what the future trajectory looks like.
Written down, an initial-value problem for an ordinary differential equation (ODE) is y'(t) = f(t, y), with y(t0) = y0. Here y is the unknown function (it may be a single number or a whole vector of them), y' = dy/dt is its rate of change, f is the given law that says how fast y changes when you are at time t in state y, and y0 is the known value at the starting time t0. The slope field f does NOT hand you y directly; it tells you the direction at each point, and the solution is the unique curve through (t0, y0) that follows those directions everywhere. For example, y' = y with y(0) = 1 has the solution y(t) = e^t, because the function whose rate of change equals itself, starting at 1, is exactly the exponential.
This is the central object of numerical ODE solving, and it matters because almost every dynamical model in science is one: planetary orbits, chemical reaction rates, population growth, the spread of an epidemic, the motion of a robot arm. A clean theorem (Picard-Lindelof) guarantees that if f is reasonably smooth in y near the start, a unique solution exists for at least a short time. But 'exists' rarely means 'has a tidy formula' — most useful f have no solution you can write with elementary functions, which is exactly why we MARCH the solution forward numerically instead of solving it on paper.
The decay law y' = -2y, y(0) = 5 has the exact solution y(t) = 5 e^(-2t): the state starts at 5 and the slope at any moment is -2 times the current value, so it shrinks faster when large and slower when small, easing toward 0. A solver never needs the formula — it only ever calls f(t, y) = -2y to get the local slope.
An IVP gives the slope everywhere plus one starting point; the solution is the curve that obeys both.
Existence and uniqueness are not free: if f is too rough in y, a solution may not be unique (y' = sqrt(|y|), y(0) = 0 has many), or it may blow up in finite time (y' = y^2 reaches infinity at t = 1/y0). A numerical solver cannot rescue a problem that is ill-posed to begin with.