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

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 conditions.

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Cite this as:

Weisstein, Eric W. "Cauchy Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchyProblem.html

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