An Improved Method of Undetermined Coefficients

ABSTRACT

An exact solution method in terms of an infinite power series is developed for linear ordinary differential equations with polynomial coefficients. The method is general and applicable to a wide range of equations of any N-th order presented in normal form. The final solution is defined by a linear combination of S functions f_j(x) j=1,...,S expressed in the form of a power series, and by additional (S-N) number of accompanied relationships for unknown constants. Each term of the series f_j(x) is defined by a finite number of operations involving matrix calculations. The term calculation is independent on unknown constants.

The method is also applicable to any system of R differential equations in normal form with the restriction that all equations in the system are of the same order. The advantage over the "classical" method of undetermined coefficients is that here the recursions are of first order even for higher-order differential equations. The general solution is expressed explicitly in the form allowing for application of any form of initial, boundary, or combined conditions. The paper presents the development of the method and, as examples, its application to solving selected second- and third-order differential equations.

Keywords:

Hermite equation, method of undetermined coefficients, polynomial coefficients, power series solutions.