An initial value problem is a differential equation
- with where is an open set,
together with a point in the domain of ƒ
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
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.
More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.
Read more about Initial Value Problem: Existence and Uniqueness of Solutions, Exponential Smoothing, Examples
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