General Solutions of Some Complex Third-order Differential Equations

According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs with entire coefficient in the neighborhood of z0.


Introduction
Ordinary differential equations(ODEs) in the complex domain is an area of mathematics admitting several ways of approach, which basic results can be found in a large number of text-books of differential equations, see, e.g. [12,13,16]. At present, many researches focus our interest on Nevanlinna theory, and have undertaken the value distribution of meromorphic solutions of ODEs, see, e.g. [2-8, 11,13-15, 17-18].
However, finding explicit solutions of ODEs in terms of finite combinations of known functions is more difficult. However, it was observed in the late nineteenth and early twentieth centuries that ODEs whose general solutions are meromorphic appear to be integrable in that they can be solved explicitly or they are the compatibility conditions of certain types of linear problems [1]. The condition that the general solution is meromorphic can be replaced by the condition that the ODE possesses the Painlvé property, that is, all solutions are single-valued about all movable singularities.
Finite order functions have special properties and so they have been the subject of intense study [10]. The major result concerning the order of growth of meromorphic solutions of first order ODEs is the following theorem due to Gol'dberg.
Theorem 1.1. [6] All meromorphic solutions of the first order ODE where Ω is polynomial in all its arguments, are of finite order. A generalization of Gol'dberg's result to second order algebraic equations have been conjectured by Bank [2]. Hayman [9] described a further generalization of Bank's conjecture to nth-order ODEs. If f (z) is a meromorphic solution of Ω(z, f, f , · · · , f (n) ) = 0, where Ω is polynomial in z, f, f , · · · , f (n) , then we have T (r, f ) < a exp n−1 (br c ), 0 ≤ r < +∞, where a, b and c are constants and exp j (x) is defined by In this paper, we will focus our interest on finding explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order differential equations.
The remainder of the paper is organized as follows. In Section 2, we recalled some results on the existence of global meromorphic solutions of second-order ODE which resulted in the characterzation of Schwarzian derivatives. In Section 3, the explicit solutions of differential equation in terms of finite combinations of known functions to solve some third-order ODEs with entire coefficient h(z) in the neighborhood of z 0 have been arrived.

Explicit Solutions of Second Order Differential Equation
When considering the formal form of second order differential equation where A(z) is meromorphic, we need first to find out whether its meromorphic solutions exist or not. The existence of global meromorphic solutions of (4) can be settled, resulting in the characterzation of Schwarzian derivatives, see Theorem 2.1 and Corllary 2.2 obtained by Herold [12]. Theorem 2.1. [12] Let G ⊂ C be a simply connected domain, such that A(z) is meromorphic in G. The quotient of any two local solutions of (4) is meromorphic and admits a meromorphic continuation into the whole G if and only if at all poles of A(z), the Laurent expansion of A(z) around z 0 has the form where and Moreover, these continuations g of local quotients all satisfy, in G, Corollary 2.1. [12] Let G ⊂ C be a simple connected domain such that A(z) is meromorphic in G. The differential equation (4) admits two linearly independent meromorphic solutions in G if and only if at all poles z 0 of A(z), the Laurent expansion of A(z) is of the form (5), satisfying (6) with an odd integer m ≥ 3 and (7).

General Solutions of Third Order Differential Equation
In this section, we discuss about linearly independent solutions of the following third order differential equation.
where h(z) is analytic |z − z 0 | < R. We want to find explicit solutions of linear differential equation (10) in terms of finite combinations of known functions, and obtain Theorem 3.1. Suppose h(z) is analytic |z − z 0 | < R, and consider the differential equation (10) in the disc |z − z 0 | < R. Let ρ 1 , ρ 2 , ρ 3 be the roots of Then (10) admits, in some slit disc D = D(r), r ≤ R, three linearly independent solutions f 1 , f 2 , f 3 of one of forms: and where Φ(z) is analytic in D.
The idea of the proof is to submit the Laurent series of f (z) and h(z) to (10) and to compare with their coefficients. By this way, we can conclude the indicial equation ρ(ρ − 1)(ρ − 2) + h(z 0 ) = 0. Theorem 3.1 shows the finite combinations of known functions f 1 , f 2 and f 3 when h(z 0 ) = 0. If h(z 0 ) = 0, we further obtain Theorem 3.2. Suppose h(z) is analytic |z − z 0 | < R, and consider the differential equation (10) in the disc |z − z 0 | < R. Let ρ 1 , ρ 2 , ρ 3 be the roots of assuming that ρ i − ρ j ∈ Z\{0}, 1 ≤ i < j ≤ 3, and h(z 0 ) = 0. Then except the forms of (11),(12), (10) also admits , in some slit disc D = D(r), r ≤ R, three linearly independent solutions f 1 , f 2 , f 3 of one of forms: where Φ(z) and φ j (z), j = 1, 2, · · · , 12 are analytic. We now give some Lemmas to prove theorems. The general solutions of differential equation come from the finite combinations of known functions. The number and forms of known functions can detect the forms of solutions. If two known functions are determinate, we have Lemma 3.1. Suppose that (10) possesses two linearly meromorphic solutions Then another solution of (10) is of the form where Φ(z) is analytic.
Proof. Assume that f = f 1 F is a solution of (10). Then Substituting the above equations into (10), we obtain where g = F .
In order to get f 3 , we need to solve the equation (20). Since f 2 is also a solution of (10),we can calculate that g 1 = f2 f1 is one solution of (20). Assume again that g = g 1 G is one solution of (20). Then we have Substituting the above equations into (20), we obtain where W = G . Solve the equation (21) and we have W = cg −2 (20). What's more, let f 3 = f 1 g 2 dz and then f 3 is the solution of (10) that is arrived.
Proof. Using the same method as in Lemma 3.3, we still need to solve equation (20), while in this case g(z) is unknown. We need to find out a set of linearly independent solutions g 1 and g 2 . Then let f 2 = f 1 g 1 dz, f 3 = f 1 g 2 dz and such f 2 ,f 3 are the solutions of (10). Assume that g In the following, we will split our proofs into six cases. Case 1. Suppose that ρ 1 = 0, 1 and k = 0, 1. Then
When ρ 1 = 2, k = −2, for any n ∈ Z, We now affirm that the formal power series If we can prove that for some r ∈ (0, R) and some M > 0,|c i |r i ≤ M holds for i = 0, 1, 2, ..., we have lim sup |c i | 1 i ≤ 1 r and therefore g 1 (z) converges. Actually, suppose that there exists some r > 0, M > 0 such that |c i |r i ≤ M for i = 0, 1, ..., n − 1.
Since (z − z 0 ) ρ1 ∞ i=1 a i (z − z 0 ) i converges and vanishes at z = z 0 , decreasing r if needed, we have for each n, ∞ i=0 |a i |r i ≤ (n+1)|a0| 7(n+2) 2 . Then for i = n, Hence g 1 = (z − z 0 ) −2 ∞ i=0 c i (z − z 0 ) i converges and is one solution of (20). Using the same method we can prove that when also a solution of (20). Therefore, we obtain the other solutions of (10) as follows: Therefore k = −2.
In this case i is a solution of (20). By Lemma 3.3, another solution of (20) is Hence, we have three linearly dependent solutions of (10) as follows: Case 6. Suppose that 1 = 0 and k = 0. Then by comparing the coefficient of the term (z − z 0 ) n in (20), we obtain the form of common term Here c 0 , c 1 is determined arbitrarily.
Based on the above lemmas, we can prove Theorem 3.1 and 3.2.
converges as a solution of (10). However if (23) doesn't hold, we cannot find out the form of f 2 in this way.
converges as a solution of (10). However if either (24) or (25) does not hold, we cannot find out the form of f 3 in this way. Thus, we need split our proofs into three cases.
Proof. of Theorem 3.2. Using the smilar method as in Theorem 3.1, except Case i and Case ii hold, we further deduce from Lemm 3.4 that one of the forms of f 1 , f 2 , f 3 as (13)− (18) holds. In this case, h(z 0 ) = 0, none of (23) , (24),(25) holds, and f 1 is the only known solution of (10).
For a special case, we also obtain Theorem 3.3. Suppose h(z) is analytic |z − z 0 | < R, and consider the differential equation (10) in the disc |z − z 0 | < R.
Then (10)admits three linearly independent solutions f 1 , f 2 , f 3 of the following forms Proof. Assume that ρ 1 > ρ 3 . Similarly as in the proof of Theorem 3.1,

By Lemma 3.3, we have
If c * k β 1 + ... + c 0 β k = 0, which means that we cannot find out the form of f 2 in this way. As h(z 0 ) = 0, by Lemma 3.4, we cannot find the solutions of (10).

Conclusion and Further Discussion
It is well known that every holomorphic function on a simply connected domain in the complex plane can be realized as the Schwarzian derivative of a function that is meromorphic on a given domain. Furthermore, This function is essentially unique by a Möbius transformation. Thus, various results about solutions to second order differential equations with meromorphic coefficients are related to this theme.
In this paper, our main result are concerned with a very particular type of a third order differential equation (10). We use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs (10). Thus, the explicit solutions of differential equation (10) in terms of finite combinations of known functions.
Throughout our paper, our results are raised from a very natural question. Some profound questions should be further discussed. Second order ODE of (8) has connection to Teichmuller theory. But, when n is greater than or equal to 3, we do not know whether there is connections with Teichmuller theory or not. Similar results hold if we take an n-th order differential equations of the same type to (10). It is more complicated for us to detect all linearly independent solutions of some n-order ODEs by using local series methods and reduction of order. We need to use computer technology on a large scale.