the initial value problem

A differential equation alone usually has too many solutions — a whole family — and in any real situation you want just one: the one that matches where things actually start. To get it, you combine the equation with a starting fact. 'The population grows at this rate, AND right now it is 1000.' That pairing — a rule plus a starting point — is an initial value problem.

Precisely, an initial value problem consists of a differential equation together with enough initial conditions, given at a single value of the independent variable, to pin down the arbitrary constants. For a first-order equation you supply one condition: solve y' = f(x, y) with y(x0) = y0. For a second-order equation you supply two, typically the value and the slope at the start: y(x0) = y0 and y'(x0) = y1 — like releasing a pendulum from a known position with a known initial push. The number of conditions matches the order, because that is exactly how many constants the general solution carries.

The phrase 'initial' reflects the physical picture: all the conditions are imposed at one instant, the start, and the equation then propagates the solution forward (and backward). Geometrically, the equation gives the family of integral curves and the initial condition selects the single curve through the starting point. This is the form most applications take — you know the law and the present state, and you want the future — which is why initial value problems, rather than bare equations, are what we usually solve. (When conditions are instead specified at two different points, it is a boundary value problem, a different beast.)