Initial value problem  From Wikipedia, the free encyclopedia
Jump to: navigation, search    An initial value problem is a
differential equation  y'(t) = f(t, y(t)) \, with  f: \Omega
\subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where
\Omega \, is an open set of \mathbb{R} \times \mathbb{R}^n,
together with a point in the domain of ƒ  (t_0, y_0) \in \Omega,
called the initial condition.  A solution to an initial value
problem is a function y that is a solution to the differential
equation and satisfies  y(t_0) = y_0. \,  In higher dimensions,
the differential equation is replaced with a family of equations
y_i'(t)=f_i(t, y_1(t), y_2(t), ...), and y(t) is viewed as the
vector (y_1(t), ..., y_n(t)). More generally, the unknown
function y can take values on infinite dimensional spaces, such
as Banach spaces or spaces of distributions.  Initial value
problems are extended to higher orders by treating the
derivatives in the same way as an independent function, e.g.
y''(t)=f(t,y(t),y'(t)).  Existence an