Modified Solutions of Linear Differential Equations with Polynomial Coeffcients near the Origin and Infinity

Linear differential equations with polynomial coeffcients are studied. Solutions near the origin and infinity are presented for the differential equations of the second order and with two blocks of classified terms, where the solutions u(t) near the origin and infinity are assumed to be expressed by a power series of t and t -1 respectively, multiplied by a power of t. In the present study, it is shown that the function which is obtained from any of these solutions by multiplying et or e/t or (l - t/), is a solution of a differential equation with two or three blocks of classified terms, where and are constants. Discussions are given also of multipliers et2 or e/t2. The studies are mainly made for the cases in which the singularities of the differential equation do not change, but some studies are given for the cases when the singularities change.