Cauchy Problem

If $f(x,y)$ is an Analytic Function in a Neighborhood of the point $(x_0, y_0)$ (i.e., it can be expanded in a series of Nonnegative Integer Powers of $(x-x_0)$ and $(y-y_0)$), find a solution $y(x)$ of the Differential Equation

$${dy\over dx}=f(x),$$

with initial conditions $y=y_0$ and $x=x_0$. The existence and uniqueness of the solution were proven by Cauchy and Kovalevskaya in the Cauchy-Kovalevskaya Theorem. The Cauchy problem amounts to determining the shape of the boundary and type of equation which yield unique and reasonable solutions for the Cauchy Boundary Conditions.

See also Cauchy Boundary Conditions