A Study on the Solution of Linear Differential Equations with Polynomial Coefficients

A linear differential equation with polynomial coefficients, which is expressed by sym.JPG is studied, where ak,m are constants. In the present study, the lefthand side of the equation is rewritten sym1.JPG where sym2.JPG and each of sym3.JPG 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.