Cauchy Problem

If  is an analytic function in a neighborhood of the point  (i.e., it can be expanded in a series ofnonnegative 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.