Initial value problem

In mathematics, in the field of differential equations, an initial value problem (also called the Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time.

An initial value problem is a differential equation

together with a point in the domain of ${\displaystyle f}$

called the initial condition.

A solution to an initial value problem is a function ${\displaystyle y}$ that is a solution to the differential equation and satisfies

In higher dimensions, the differential equation is replaced with a family of equations ${\displaystyle y_{i}'(t)=f_{i}(t,y_{1}(t),y_{2}(t),\dotsc )}$, and ${\displaystyle y(t)}$ is viewed as the vector ${\displaystyle (y_{1}(t),\dotsc ,y_{n}(t))}$. More generally, the unknown function ${\displaystyle y}$ can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.

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