initial condition
If a differential equation describes a family of possible histories, an initial condition is the snapshot that says which history is actually yours. It is the single piece of present knowledge — where you are right now — that, combined with the rule for how things change, determines the whole story. Without it the equation knows the shape of every possible future but not which one you live in.
Precisely, an initial condition specifies the value of the unknown function (and, for higher order, of its derivatives) at one particular value of the independent variable, such as y(0) = 5 or y'(0) = 2. Each condition lets you solve for one arbitrary constant: substitute the given point into the general solution and solve. A first-order equation needs one initial condition; an nth-order equation needs n of them, all at the same point — one for the function and one for each derivative up to order n - 1. Together they fix every constant and select a unique solution.
Geometrically, an initial condition is a single point in the plane that the solution curve must pass through, and the equation's slope field then determines the rest of the curve from that anchor. This is why a tiny amount of information — one or two numbers — is enough to lock down an entire function: the differential equation supplies all the structure, and the initial condition only has to choose among the family. In modelling, the initial condition is simply the measured present state, the thing you actually know before you predict the future.
For the second-order equation y'' + y = 0 with general solution y = C1 cos(x) + C2 sin(x), the two initial conditions y(0) = 1 and y'(0) = 0 give C1 = 1 and C2 = 0, so the unique solution is y = cos(x).
n conditions at one point fix n constants — one per derivative up to order n - 1.
All initial conditions must be given at the SAME point of the independent variable. Conditions split between two points make it a boundary value problem instead, which need not have a unique solution — or any.