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.

See also

Cauchy Conditions, Cauchy-Kovalevskaya Theorem

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Cite this as:

Weisstein, Eric W. "Cauchy Problem." From MathWorld--A Wolfram Web Resource.

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