### Formal solutions of differential equations

Michael. F Singer
1990 Journal of symbolic computation
We give a survey of some methods for finding formal solutions of differentim equations. These include methods for finding power series solutions, elementary and liouvillian solutions, first integrals, Lie theoretic methods, transform methods, asymptotic methods. A brief discussion of difference equations is also included. In this paper, I shall discuss the problem of finding formal expressions that represent solutions of differential equations. By using the term "formal", I wish to emphasize
more » ... fact that most of the time I will not be concerned with questions of where power series converge or in what domains the expressions represent functions. I shall talk about power series solutions, solutions that can be expressed in terms of special functions such as exponentials, logarithms, or error functions, solutions given implicitly in terms of elementary first integrals and Lie theoretic techniques. I shall briefly mention transform methods, asymptotic expar~sions and devote a final section to a short discussion of formal solutions of difference equations. There are many open problems in these areas and I have included my favorite ones. I hope they will stimulate further work. I would like to thank Bob Caviness, Leonard Lipshitz, Marvin Tretkoff, and the referees for helpful comments on an earlier version of this paper. I. P o w e r Series Solutions of Differential E q u a t i o n s My aim here is to contrast what is known about linear differential equations with what is known about non-linear differential equations. Good general references for information about linear differential equations are Poole (1960) and Schlesinger (1895). Consider the linear differential equation z(y) = an_l(x)y/n-1) + . . . + a0(x)Y = 0 This is an e x p a n d e d and revised version of talks presented at