Families of principal solutions of ordinary differential equations.
Erling William Chamberlain · Transactions of the American Mathematical Society · 1963
IntroductionLet A(y, Dy,--,D"y) = 0 be an algebraic differential equation belonging to Strodt's class (D) [1, p. 5], and let M be a principal monomial [1, §66] for A = 0.In search for principal solutions of A = 0 (i.e., solutions which are ~ M) one substitutes y = M(l + z).This almost always [2, §121] leads to a differential equation P(z) = 0 in which F is an asymptotically quasilinear algebraic differential operator having a nonexceptional factorization sequence (Wx, •■-, Wn) such that F is normal with respect to (Wx,---,W",r) for a sufficiently large positive integer r.Strodt [2] has shown that F = 0 has at least one solution Z -) denoting the direction in which the complex variable approaches infinity).The significance of the number « in the more general a.q.l.situation has hitherto been obscured by the complicated manipulations used to ascertain the mere existence of a solution -< 1 for a.q.l.equations.In the present study, a «-parameter family of solutions -< 1 is exhibited for a large class of normal a.q.l.equations P = 0. We require that the factorization sequence for P satisfy a condition which resembles (and includes) the classical "distinctness of characteristic roots" in the case of linear differential equations with constant coefficients ( §2, below).The virtue of this condition is that it allows us to pass from approximate factorizations of linear operators in terms of the operators Wi = (1 -W¡~ 1D) to exact factorizations in terms of certain Vt such that Vt ~ W¡.Once the linear part of an algebraic a.q.l.equation has been exactly factored, the situation may be said to be well in hand.Among other things, exact factorization enables us to measure with considerable precision the asymptotic size of our small (^ 1) solutions, and even to obtain a sort of asymptotic development for them.Other phenomena used to advantage in this study include the invariance of normality and asymptotic quasilinearity of differential operators