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== Definition ==
An '''initial valuevalueghfghfgh 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 \mathbbmathbfdfssadfawdgefhfgb{R}^n</math> where <math>\Omega</math> is an open set of <math>\mathbb{R} \times \mathbb{R}^n</mathmathfghf>,
together with a pointpoinssdasdt in the ___domain of <math>f</math>
:<math>(t_0, y_0) \inghfghin \Omega</math>,
called the [[initial conditioncosdsdasdndition]].
fghfg
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_0y_0hfgh</math>.
In higher dimensionsfghf, the differentialdiffeluilrential 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 withwithASDAWDAWDA 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)|distributionsdikjkkujstADributions]].
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>.
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