On the Equivalence of Differential Equations in the Sense of Coincidence Reflecting Functions

Abstract

We use a new method for constructing some differential equations which are equivalent to a given equation in the sense of having the same reflecting function. We completely solve the problem: when is a polynomial differential equation equivalent to a given polynomial differential equation? Many sufficient conditions have been established for one differential equation to be equivalent to a given differential equation. We apply the obtained results to study the boundary value problem of two equivalent differential equations.

1. Introduction

By [1, 2] we know that some polynomial differential system where and are polynomial functions, can be transformed to an equation of the form where are polynomial in , . The fact that systems with a homogeneous nonlinearity can be transformed to (2) with has been exploited in a number of previous papers [1–4]. The limit cycles of (1) correspond to -periodic solutions of (2). This fact has been used to facilitate certain computations and to provide some information about the global phase portrait of the system. In this paper we will use the method of Mironenko [5] to construct some differential equations which are equivalent to (2) in the sense of having the same reflecting function.

Now, we simply introduce the concept of the reflecting function.

Consider differential system which has a continuously differentiable right-hand side and general solution . For each such system, the reflecting function is defined as [5].

If system (3) is -periodic with respect to , then is the Poincaré mapping of (3) over the period . Thus, the solution of (3) defined on is -periodic if and only if is a fixed point of .

A differentiable function is a reflecting function of system (3) if and only if it is a solution of the Cauchy problem

If the reflecting functions of two differential systems coincide in their common domain, then these systems are said to be equivalent [5].

If is the reflecting function of (3), then it is also the reflecting function of any system where is an arbitrary vector function such that the solutions of every system above are uniquely determined by its initial conditions. The systems (3) and (5) have the same operators of translation along solutions on the symmetric time interval , and therefore the initial data of solutions of the boundary value problems of the form , where is an arbitrary function, coincide for such systems.

In general, it is very difficult to find out the reflecting function of (3), so to writ out system (5) is difficult, too. How to judge if two systems are equivalent when we do not know their reflecting function? This is a very interesting problem! Mironenko in [5–7] has studied it and obtained some good results.

Lemma 1 (see [6]). If continuously differentiable vector functions are solutions of differential system then all perturbed systems of the form where are arbitrary continuous scalar odd functions, are equivalent to each other and to system (3).

The theory of reflecting function has been used for studying the qualitative behavior of solutions of differential systems by many authors [5–13].

Now, for a given equation (2), we present a method for constructing other first-order differential equations which are equivalent to (2), that is, to find out the solution of (6) with . However, it is impossible to find all the solutions in most cases. Therefore, we take only polynomial solutions of (6), that is, its solutions of the form where the coefficients are assumed to be differentiable functions on and is not to be zero identically.

Bel’skii in papers [8–10] has discussed, respectively, when a first-order polynomial differential equation is equivalent to a linear equation and Riccati equation () and Abel equation (). In this paper, we will in detail discuss when is a first-order equation equivalent to (2) with . We have obtained the sufficient conditions for (6) has solution in form of (8) and we give the explicit expressions of . At the same time, in addition, we also find out some nonpolynomial solutions of (6). The results of this paper generalize the results of Bel’skii [8–10], and the method of proof in this paper is more simple and efficient than in the previous papers.

2. Main Results

Now, we consider (2) and assume that is not to be zero identically in and . By Bel’skii [8], we know that if , then the degree of polynomial is equal to . In the following, we suppose that and .

Theorem 2. For (2), suppose that and . Then the function of the form (8) is a solution of (6), if and only if Therefore, can be represented in the form

Proof. Let us substitute the function of the form (8) into (6); we get Equating the coefficients of the like power of , we obtain where and , when .
From (16) when we get . Taking in (16), we have . Similarly for , , we get that is, relation (11) is true.
Substituting (11) into (16) and simplifying we get (10).
Using relation (11) and taking in (16), we get it implies relation (12).
Using relation (12) and taking in (16), we have substituting into the above, we get that relation (13) is held.
Substituting (11)–(13) into (8), we get which completes the proof.

We rewrite as follows: in which

Lemma 3. If , , , then

Proof. As , , so (21) goes to equating the coefficients of and ; from here we get relations (23) and (24).

Theorem 4. Suppose that , , , and where Then the function in (8) is a solution of (6); that is, (2) is equivalent to equation where is an arbitrary continuous odd function.

Proof. For , by Lemma 3, substituting (12) and (13) into (10), we obtain Substituting (24) into this relation, we obtain By (11) and (23), we get As , (37) is an identity when . For , solving (37), we have Substituting (38) into (11)–(13), we obtain (29)–(32).
On the other hand, by relation (10), for , we have Substituting (29)–(32) and (37) into the above, we obtain which deduces that relation (26) is true. By Theorem 2 and in (10) taking we get that relations (27) and (28) are held.
By Lemma 1 and Theorem 2, the proof is completed.

Theorem 5. Supposing that , where where and are arbitrary constants. Then the function of the form (8) is a solution of (6); that is, (2) is equivalent to (34).

Proof. As , and , by Lemma 3, we have Using , it is not difficult to check that (37) is an identity when . For , in relation (10) taking and computing, it follows that which implies that Combining relation (12), it yields solving this equation we get the expression (45).
Substituting (49) into (13) implies (42). Substituting (42)–(45) into equation and simplifying we get . By Theorem 2, the proof is finished.

Similarly, we get the following.

Theorem 6. Supposing that , and , where and are arbitrary constants. Then (2) is equivalent to (34).

Theorem 7. Supposing that , , where are arbitrary constants. Then (2) is equivalent to (34).

By the literature [5], we know the following.

Corollary 8. If all the conditions of one of the above four theorems (Theorems 4–7) are satisfied and (2) is -periodic in , then the initial data of solutions of the boundary value problems of the form , where is an arbitrary function, coincide for systems (2) and (34).

Example 9. The equation has the only one -periodic solution . For this equation, we have By Theorem 6, we get where and are constants: As then that is, the first condition of Theorem 6 is satisfied. Thus Taking and , we get Taking and , we have Therefore, (54) is equivalent to equation where are arbitrary continuously odd functions. By Corollary 8, (63) has only one solution such that .

Now, taking in the above theorems, we get the following corollaries.

Corollary 10. Suppose that ; then where and are arbitrary constants; . Then the Abel equation is equivalent to equation where is an arbitrary continuously odd function.

Corollary 11. Suppose that , , and ; where and are arbitrary constants. Then Abel equation (66) is equivalent to (67).

Corollary 12. Suppose that , , and ; where are arbitrary constants. Then Abel equation (66) is equivalent to (67).

Example 13. Abel equation has three constant solutions , , and . For this equation, we have , , , , and .
Thus and .
Solving , we get where , are arbitrary constants. Therefore Taking and , we obtain Taking and , we get Consequently, by Lemma 1 and Corollary 10, Abel equation (72) is equivalent to equation where are arbitrary continuously odd functions. By Corollary 8, (78) has only three solutions such that ( is a nonzero constant).

Remark 14. The conclusions of the above three corollaries have been proven by Bel’skii and Mironenko in [9].

Because (6) is a linear equation, if are solutions of (6), then also is solution of (6), where are arbitrary constants. In the following, we will find out some solutions of (6) in the other form.

Theorem 15. For the first-order differential equation , if are solutions of (6), then , , is a solution of (6), too.

Proof. As and , Thus, the proof is completed.

By Example 9 and Theorem 15 we get that equation is also equivalent to equation where are arbitrary continuously odd functions, , are arbitrary constants, and are the same as in Example 9. This equation has only one solution such that .

Using Theorem 15 and Corollary 10 we get the following.

Corollary 16. Supposing that , Then Abel equation is equivalent to equation where are arbitrary continuously odd functions and is a constant: in which , are arbitrary constants, and . Consider

Theorem 17. For equation is a solution of (6). Thus, (89) is equivalent to equation where is an arbitrary continuously odd function, is an arbitrary differentiable function, and is a continuous function.
This result is easy to be proven.

Obviously, from Theorem 17, we see that if is a polynomial of , then the corresponding equation (6) has at least one polynomial solution . This is implied by taking in (90).

By Theorem 17, (72) is also equivalent to If in (92) we put , then we obtain where is an arbitrary continuously odd function. This equation has only three solutions such that ( is a nonzero constant).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the NSF of Jiangsu of China under Grant no. BK2012682 and the NSF of China under Grant nos. 11271026 and 61374010.