Cauchy Problem

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

with initial conditions and . 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.