Initial value problem

Definition

Let $\Omega$ be an open subset of $\mathbb{R} \times \mathbb{R}^m$, let $f : \Omega \to \mathbb{R}^m$ be a function, and let $(t_0, y_0) \in \Omega$. The initial value problem for $f$ at $(t_0, y_0)$ is the system $$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.$$ A solution of the problem is a solution $y : I \to \mathbb{R}^m$ of the equation $y' = f(t, y)$ on an open interval $I$ with $t_0 \in I$, such that $y(t_0) = y_0$. The condition $y(t_0) = y_0$ is the initial condition.