Initial value problem

A differential equation plus a known starting point, together pinning down ONE specific solution.

An initial value problem (IVP) pairs a differential equation with an INITIAL CONDITION — a known value $y(x_0)=y_0$ — which pins the arbitrary constant in the general solution down to one specific, PARTICULAR solution. Without the initial condition, a differential equation typically has infinitely many solutions (one per value of $C$ ); the initial condition selects exactly one.

Examples

For $y'=2x$ , $y(0)=5$ : the general solution $y=x^2+C$ becomes $y=x^2+5$ once $C$ is solved from $0^2+C=5$ .