Initial-value problem for an ODE

An initial-value problem prescribes

y'=F(t,y),\qquad y(t_0)=y_0.

For continuous $F$ , a classical solution is equivalently a continuous function satisfying the integral equation

y(t)=y_0+\int_{t_0}^tF(s,y(s))\,ds.

The equivalence follows from the fundamental theorem of calculus. Existence and uniqueness require additional hypotheses on $F$ .

Integration constants

For $y'=g(t)$ , antiderivatives differ by an integration constant; the initial value fixes it. In a parameterized problem, the initial value and hence this constant may depend on the parameter. Prescribing data at two endpoints is a boundary-value problem and has different solvability conditions.