Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind Ab-1 equations . Based on the results of a previous work, concerning a closed-form solution of a general Ab-1 equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals in closed or parametric form have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.


Introduction
Autonomous equations, as it is well known, often arise in mechanics, physics, and chemical engineering since a considerable number of problems are governed by weakly or strongly nonlinear equations of this kind.For example in the study of damped oscillators one records famous equations, extensively investigated in the literature, like the ones governing the Duffing 1 or the Van der Pol 2 oscillator.Both these equations are of the Liénard type: where k and m are differentiable functions of y.Furthermore, 1.1 as well as other autonomous equations like the Rayleigh 3  with l a differentiable function of q.Equations 1.1 and 1.2 have been studied thoroughly in the literature with regard to the stability of their critical points and the number of the limit cycles which correspond to global or local bifurcations, especially in the case where k, l, m are polynomials.For example, we refer to 5-9 as far as 1.1 is concerned and 4, 10, 11 for the more generalized case 1.2 .We should also note that a lot of authors obtain exact or approximate first integrals also called adiabatic invariants .See for instance the work of Kooij and Christopher for the integrability of planar polynomial systems by means of algebraic invariant curves 12 , as well as the works of Denman 13 and Van Horssen 14 , where approximate invariants are obtained via perturbation techniques.Moreover, numerous works based on various perturbation methods yield approximate solutions and or qualitative results see the book of Verhulst 15 and the references there in .In addition, some series solutions have been derived in the literature, concerning nonlinear ODEs, where the developed methods can be applied in general to cases of autonomous equations.For example, we refer to functional analytic techniques resulting in this kind of solutions see 16, 17 .However, not much progress has been made as regards the derivation of exact, general, closed-form solutions of the equations studied in the above references.Thus by considering autonomous equations of a polynomial structure for dy/dx up to the second degree , with coefficients of a not necessarily polynomial form for y see, e.g., the Langmuir equation 18 , in the present work we investigate analytically this generalized polynomial form, aiming at the construction of proper techniques, capable of removing the difficulties arising in the derivation of exact solutions.Most of the above mentioned equations are presented by Davis 19, Chapter 7, Section 2 .A significant part of the relevant search in procedures of this kind deals with the use of appropriate transformations.However, the classic transformation y x q y , usually applied to autonomous nonlinear ordinary differential equations of the second order results in Abel equations of the second kind see, e.g., 20, Section 2.2.3 , which in general cannot be solved analytically, except in special cases, most of which accept only parametric solutions see 20, Sections 1.3.1-1.3.4 therefore a parametric solution for y, y x is derived as regards the considered autonomous equation .Hence, in Section 2, in order to construct a more efficient analytical technique, concerning two general subclasses of the autonomous equations under consideration, we use another, properly modified, general transformation, to obtain Abel equations of the first kind.Furthermore in 21 , an implicit solution of a general Ab-1 equation has been obtained, together with the associated sufficient condition.
Then in Section 3, introduction of an arbitrary function in combination with the derived in 21 solution, yield one-parameter families of solutions for the original nonlinear autonomous equation.More specifically, by means of the sufficient condition being extracted for the solution of the Abel equation, the arbitrary function is determined so that a first integral of the autonomous equation of the form dy/dx h y is to be derived.As application, we obtain families of solutions concerning four specific cases of Liénard equations, for which first integrals in parametric form or parametric solutions have been derived so far.Finally, in Section 4, by using the solutions extracted in Section 3, we conclude solutions for families of nonlinear autonomous systems of differential equations of the first order, equivalent to the equations considered in this work.Two examples of such families International Journal of Differential Equations 3 are given, where solutions are obtained in combination with solvable cases of autonomous equations, presented in Section 3.

Reduction of a General Autonomous Equation
Hereafter the prime denotes differentiation with respect to the corresponding suffix.We consider the following general autonomous differential equation of the second order: with g i y , i 0, . . ., n, continuous functions of y.In particular for n 1 we have the Liénard equation.By applying the transformation M : we arrive at the Ab-1 equation:

2.3b
We further consider the Abel equation of the first kind: Then in 21 , by taking into account a solvable in closed-form Abel equation 20, Section 1.4.1.47c −a/b , involving arbitrary functions and using a transformation given by Kamke 22, Chapter A, Equation 4.10.d, we have finally proved the following theorem.
Theorem 2.1.If the following relation holds: then the Abel equation 2.4 f 2 / 0 has a general implicit solution of the form where c is an arbitrary parameter and K stands for the parameter of the family of solutions.
In fact, one more sufficient condition is extracted together with 2.5 see 21, Equation 3.2 , constituting a relation between the arbitrary functions involved in the auxiliary International Journal of Differential Equations "arbitrary" equation.These functions are not included in the extracted solution and hence we can claim that they are finally eliminated, "allowing" the derivation of the closed-form solution 2.6 .We also note that a similar condition is given by Kamke 22, Chapter A, Equation 4.10.f, limited to the Liénard equations, but the resulting implicit formulas have a rather more complicated structure than the equation obtained here as regards the "algebraic" evaluation of the dependent variable see 21, Equations 3.16 -3.18 .

Construction of Exact Solutions
Let us consider a first order ordinary differential equation of the general form: 3.1 By introducing an arbitrary function G x , we write the system of equations: where f is a known continuous function of y.By replacing now x with y and y x with p y , we write 3.2 and 3.3 with respect to 2.3b or 2.3a when n 1 g 2 0 , as with F 1 y, p g 0 p 3 g 1 p 2 g 2 p, F 2 y, p 0, and f p p 3 .Then, by means of Proposition 3.1, we prove the following theorem.
with G as in 3.8 .
Proof.Since 3.5 is an Ab-1 equation of the form 2.4 with y instead of x and p y instead of y x , application of the sufficient condition 2.5 where x is replaced by y and in the linear equation: Then 3.8 is obtained as the solution of 3.10 .Moreover, according to Theorem 2.1, 3.5 has a closed-form solution, given by 2.6 modified as regards the variables .On the other hand, integration of the separated variables equation 3.6 yields where B is an integration constant.
By equating now the right-hand sides of 3.5 and 3.6 , we conclude to the relation: where substitution of 3.11 for p, results in 3.9 .The parameters κ, Γ involved in the expression obtained for G 3.8 , as well as the parameter B appearing in 3.11 , can be determined by means of 3.9 , where 3.8 is substituted for G. Hence, the function G y can be determined so that 3.5 and 3.6 to have a common solution given from 3.11 or 2.6 .It follows from Proposition 3.1 that this solution satisfies 2.3b or 2.3a when n 1 and therefore constitutes a first integral of the autonomous 2.1 .Finally integration of 3.11 yields 3.7 and the proof of the theorem is complete.
Thus use of Theorem 3.2 in the case of an autonomous equation of the form 2.1 , means that we determine at first the parameters κ, Γ, B by means of 3.9 , then we obtain G y by using 3.8 , and finally by substituting B and G in 3.7 we arrive at a one-parameter family of solutions for the considered equation.We apply now this procedure to four cases of Liénard equations.The first case Example 3.3 concerns the general form of an equation the International Journal of Differential Equations solution of which can not be found in 20 .In fact, a special case of this equation is presented by Polyanin and Zaitsev, where the proposed procedure results in a parametric solution for y, y x .The same authors 20, Section 2.2.3 arrive at this kind of solution for the other two cases as well Examples 3.4 and 3.6 by means of the classic transformation y x q y , the equations are finally reduced to Abel forms of the second kind , while as regards the fourth case Example 3.5 , in 20 a parametric solution for x, y is presented.In general, as regards the parametric solutions, besides the parameter can not be eliminated except for very special cases in this case a first integral would be derived for the considered autonomous forms , it should be noted that this kind of solutions cannot be handled easily, since in many cases it is very difficult to determine the domain of validity of the parameter for the problem under consideration.Moreover, it is worth to be mentioned that any of the reduced Ab-1 equations 2.3a and 2.3b , which correspond to the considered examples, is included in the solvable cases of the Abel equations of the first kind presented in 20, Section 1.4.1 .

3.33
where κ is given by 3.32 .
We should note that in certain cases, depending on the form of g i y , i 0, 1, 2, in order to determine κ, Γ, B by means of 3.9 , we may need to determine one or more parameters of the original equation, as well, or establish appropriate relations concerning these parameters.This means that application of Theorem 3.2 yields families of solutions valid for special cases of the considered equation.Obviously, when these special cases concern "degenerate" forms, like the linearized ones or equations with g 0 0, then the developed herein analytical method becomes not appropriate for the specific autonomous equation.For example, if we consider an equation of the form y xx a 0 ay 2 y x β 0 βy γy 3 0, a / 0, 3.34 which is the general form of the reduced second-order equation corresponding to the Fitzhugh-Nagumo system with x t instead of y x , see 17 , then by 3.9 we obtain κ, Γ, B 0, 0, 0 , holding for the case where β 0 , β, γ 0, 0, 0 .On the other hand, an example where the present analysis arrives at exact solutions for special cases of the considered equation, is the following: y xx a 0 ay 2 y x β 0 βy 2 0, a / 0.

Autonomous Systems Equivalent to 2.1 and Exact Solutions
By constructing two-dimensional autonomous systems of differential equations equivalent to 2.1 , then based on Theorem 3.2 we can prove the following proposition.

Proposition 4.1. The autonomous system of ordinary differential equations of the first order:
x t − 1 g ,x g 2 y g 2 x, y g ,y g 1 y g x, y g 0 y , x x t, y x g x, y , y y t ,

4.1
where g x, y is an arbitrary function with continuous partial derivatives g ,x , g ,y , and g i y , i 0, 1, 2, g 0 g 1 / 0, continuous functions of y, has the following one-parameter family of solutions: where A represents the parameter of the family, G y is given by 3.8 and the parameters κ, Γ, B are obtained by 3.9 .
Proof.Let us consider the following system: The aim of the above "constructive" proposition is that, for every function g x, y possessing continuous partial derivatives, a family of nonlinear autonomous systems can be constructed by means of 4.1 , which can be solved exactly via 3.8 , 3.9 , and 4.2 .For example, we consider two specific forms for g, combined with solved autonomous equations, presented in Section 3.

Discussion and Conclusion
In this work we have taken advantage of transformations provided for the Abel equations of the first kind.Equations of this kind are obtained by a proper general transformation and they represent the reduced forms of two general subclasses of nonlinear autonomous equations of the second order investigated here.Further, in a previous work we have considered a specific Abel equation including arbitrary functions, which can be solved in closed-form, and using another transformation introduced by Kamke , we have finally derived a sufficient condition yielding an implicit solution of a general Ab-1 equation.We note in particular that the arbitrary functions are finally eliminated, "allowing" the derivation of this closedform solution.
Moreover in this work, another arbitrary function, introduced in the analysis, takes advantage of the extracted solution for an Ab-1 equation , yielding exact one-parameter closed-form solutions concerning the original autonomous equations.Hence we can assert that, although arbitrary, all these functions have been profitably used in the analysis developed in these works.
In conclusion, regarding the choice of appropriate analytical tools we can claim that the use of Abel equations of the first kind gains an advantage over other analytical methods as far as certain general classes of autonomous nonlinear second-order ODEs, as well as equivalent to these forms two-dimensional autonomous nonlinear systems of first-order ODEs, are concerned.

Proposition 3 . 1 .
If there exists a function G x such that 3.2 and 3.3 have a common solution, then this solution satisfies 3.1 as well.Proof.Proposition 3.1 follows easily, since by combining 3.2 and 3.3 and taking into account 3.4 , we obtain 3.1 .

Theorem 3 . 2 . 7 International
The nonlinear autonomous equation 2.1 has the following exact one-parameter families of solutions: dy B − G y dy ±x A, 3.Journal of Differential Equations 5 where A stands for the parameter of the family, G is given by G y g 1 exp g 2 dy Γ κ g 1 exp − g 2 dy dy g 0 , 3.8 and the parameters κ, Γ, B are evaluated by means of the following relation: , and f any continuous function of y.In 20 three equations of the form 3.27 are presented 20, Section 2.2.3.10 f e λy , Section 2.2.3.19 f sin λy , Section 2.2.3.20 f cos λy .Here we have that International Journal of Differential Equations with g arbitrary function with continuous partial derivatives of the first order and h another arbitrary function.We additionally consider 2.1 n 2 where x is replaced by t, namely, Thus by substituting the right-hand sides of the second equation of 4.3 and 4.5 , for y t and y tt , respectively, and solving for h x t , 4.4 concludes to the first of 4.1 .Therefore the system 4.1 is equivalent to 4.4 and hence, from Theorem 3.2 it follows that the solution for y t is given by the first equation of 4.2 , while combination of 3.11 with t instead of x with the second equation of 4.1 yields the second equation of 4.2 .The proof of the proposition is complete.
00 C 10 x C 01 y C 20 x 2 C 11 xy C 02 y 2 C 30 x 3 C 21 x 2 y C 12 xy 2 C 03 y 3 y t A 0 Ax By Lx 2 Mxy Ny 2 , 00 D 10 x D 01 y D 20 x 2 D 11 xy D 02 y 2 D 30 x 3 D 21 x 2 y D 12 xy 2 D 03 y 3 a A 0 Ax By Lx 2 Mxy Ny 2 10 x E 01 y E 20 x 2 E 11 xy E 02 y 2 e y , E ij A 0 , A, B, a, γ , i, j 0, . . ., 3, provided from the first of 4.1 .According to the derived solutions in the Example 3.6 of Section 3, 4.2 conclude to ij can easily be obtained by the first equation of 4.1 and hence it is not necessary to be given here.Taking now into account the results obtained above Section 3, Example 3.3 , for the set of parameters a, β, γ 3, 1, 1 , 4.2 ij