A remark on meromorphic solutions of differential equations

Abstract

The following assertion is proven. Suppose the functions F, P, Q of the differential equation F(z, w, w', w(',)) = P(z, w)/Q(z, w) to be polynomials of all their corresponding variables. If the considered equation has a transcendental meromorphic solution in \z\ < oo, then Q(z, w) does not depend on w. An example of possible applications is stated. We would like to recall the following problem which has been widely investigated in the analytic theory of differential equations: to find, if possible, all differential equations of a given form which have solutions of a given class; for example: to find all first order differential equations of the form>>' = f(x, y), where f(x,y) is a rational function of x and.y, and every solution of which is a transcendental meromorphic (entire) function (known complete result: the Riccati (linear) equation y' = y2 + p(x)y + q(x) (y' = p(x)y + q(x)) which alone has the required property [1], [4]). This note deals with a question of this kind. We consider the differential equation (1) F(z, w,w',..., wM) = P(z, w)/Qiz, w), where F, P and Q are polynomials of all their corresponding variables. Our main result is contained in the following assertion. Theorem. Suppose the polynomials Piz, w) and Qiz, w) to be mutually prime. If there exists a transcendental meromorphic solution of equation (1) in the finite z plane, then Qiz, w) does not depend on w. We would like to propose an application of this theorem: an equation of the form (2) win) = Piz,w)/Qiz,w) with polynomials P and Q can have a transcendental meromorphic solution only if Q iz, w) does not depend on w, that is, (2) must be of the form (3) w{n)=Piz,w) where P(z, w) is a polynomial of w and a rational function of z. Very simple considerations show that

Meromorphic solutions of algebraic differential equations

“…In [30] and [54] independent proofs are given that for the absence of an admissible solution of (6.14) it is necessary that deg Q = 0 and deg Ρ < κ(#). Various special cases of this assertion were obtained earlier in [40] and [58]. [59] contains the same result, but with a different definition of an admissible solution.…”

“…If F in Theorem 10 is linear in H, then the equation is (6.14) Ш .. 1У ) = Ш-, P to which [30], [40], [58], [59], and [54] are devoted. We assume that Ρ and Q are relatively prime.…”

On the order of growth of meromorphic solutions of first-order differential equations

“…If we set Vo(Z ) = 1/(yo(Z )-a), then since Yo satisfies (14), it easily follows that v o satisfies,…”

“…Other proofs and other generalizations of these theorems have been obtained by various authors including Wittich [17], Laine [9,10], Gackstatter and Laine [4], Hille [7], and Strelitz [14]. (For a survey of some of these results, we refer the reader to pp.…”

Zur Theorie der binomischen Differentialgleichungen

“…Fiir diesen bekannten Satz von Malmquist [7] gibt es verschiedene neuere Beweise, u.a. yon Yosida [ 15,16], Wittich [13,14], Laine [6] und Strelitz [11 ]. Von Yosida [15] stammt die folgende Verallgemeinerung des Malmquistschen Satzes:…”