Editing Initial value problem (section)

== Definition ==
An '''initial value problem''' is a differential equation
:<math>y'(t) = f(t, y(t))</math>  with  <math>f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n</math> where <math>\Omega</math> is an open set of <math>\mathbb{R} \times \mathbb{R}^n</math>,
together with a point in the domain of <math>f</math>
:<math>(t_0, y_0) \in \Omega,</math>
called the [[initial condition]].

A '''solution''' to an initial value problem is a function <math>y</math> that is a solution to the differential equation and satisfies
:<math>y(t_0) = y_0.</math>

In higher dimensions, the differential equation is replaced with a family of equations <math>y_i'(t)=f_i(t, y_1(t), y_2(t), \dotsc)</math>, and <math>y(t)</math> is viewed as the vector <math>(y_1(t), \dotsc, y_n(t))</math>, most commonly associated with the position in space. More generally, the unknown function <math>y</math> can take values on infinite dimensional spaces, such as [[Banach space]]s or spaces of [[distribution (mathematics)|distributions]].

Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. <math>y''(t)=f(t,y(t),y'(t))</math>.