A Study on the Solution of Linear Differential Equations with Polynomial Coefficients
Tohru Morita *
Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
Ken-ichi Sato
College of Engineering, Nihon University, Koriyama 963-8642, Japan
*Author to whom correspondence should be addressed.
Abstract
A linear differential equation with polynomial coefficients, which is expressed by 
is studied, where ak,m are constants. In the present study, the lefthand side of the equation is rewritten 
where 
and each of 
is called a block of classified terms in Lu(t). The solution is presented by taking advantage of the expression of the differential equation in terms of blocks of classified terms. When the differential equations is of the second order, six differential equations with two blocks of classified terms are chosen, such that their solutions are ordinarily expressed by the hypergeometric series, or the confluent hypergeometric series, or other two related series, except for some special values of coefficients. It is shown that all the other differential equations with two blocks of classified terms are reduced to one of these six by a change of variable.
Keywords: Linear differential equations with polynomial coefficients, Laplace's differential equation, AC-Laplace transform, change of variables