Initial value problem

In mathematics, an initial value problem is a statement of an ordinary differential equation together with specified value of the unknown function at a given point in the domain of the solution. Calling the given point t₀ and the specified value y₀, the initial value problem is

$y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}$

The problem is then to find y(t) for arbitrary t.

This statement subsumes problems of higher order, by interpreting y as a vector. For derivatives of second or higher order, new variables (elements of the vector y) are introduced.

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