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 conditions.