Research Article Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

Transcription

1 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 007, Article ID 5969, 5 pages doi:0.55/007/5969 Research Article Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs M. P. Markakis Received March 007; Revised 9 May 007; Accepted 5 June 007 Recommended by Martin J. Bohner We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration. Copyright 007 M. P. Markakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.. Introduction The class of equations solved by elliptic functions, like.) wefirstexaminehere Section ), triggered off the problem of the classification of the general nonlinear differential equation to special categories with respect to the character of the singular points of the solutions. The investigation was undertaken by E. Picard, P. Painlevé, B. Gambier, and their associates around the beginning of the 0th century, and led to the production of a large number of memoirs, listed by Davis in [,Bibliography].Wereferhereto[ 9]. The problem was also extensively presented in 97 by Ince in [0]. The French analysts mentioned above adopted in their study the polynomial form of a second-order nonlinear differential equation [, Chapter 8, Section, equation ); Section 5, equation 3)] and tried to establish conditions under which the critical points of a solution, that is to say branch points and essential singularities, would be fixed points,

2 International Journal of Mathematics and Mathematical Sciences while the functions representing the solutions of these equations would have only poles as movable singularities. Their work resulted in the discovery of 50 types of equations recorded by Davis in [, Appendix ]) possessing the above property. Of these, all but 6 are integrable in terms of classical and or) transcendental functions. The remaining six equations, known as Painlevéequationsdue to both Painlevé and Gambier), require the introduction of new transcendental functions for their solution, the so-called Painlevé transcendents. Inaddition,manynonlinearODEsofthesecond-aswellasofthefirst-order,governing various problems in Mechanics and Physics, do not accept analytical solutions in terms of known functions. As an example, we mention the nonlinear oscillator equations Rayleigh, Van der Pol, Duffing) which have been investigated thoroughly in the literature also by the present author, see []), as well as the most cases of Abel equations of the second kind the few solvable cases are presented by Polyanin and Zaitsev []). Thus, the establishment of methods leading to the construction of new functions, by means of which exact analytical solutions can be extracted, would be a desirable advance in the theory of nonlinear differential equations. In this work, aiming at this target, in Section we treat a nonlinear equation due to Gambier equation.)), which under a specific functional transformation can be reduced to a form included in the group of 50 equations mentioned above. Then, in Section 3 we consider a method leading to a more general analytical solution for this specific equation, and in Section 4 we develop a general analytical technique applicable to another nonlinear ODE equation 4.)) not included in the above list of 50 equations. This technique is based on the introduction of a new function, called, by which we obtain an exact closed form solution depending on the parameters of the equation and containing two arbitrary constants which yield a special solution in the case of an initial value problem. In Section 5 we investigate the determination of the function. Finally, in Section 6 we apply the obtained results to two special cases of 4.) equations 6.) and 6.)) and extract their exact solutions formulas, while in Section 7 we present the expressions and graphics concerning the new functions involved in the above obtained solutions. We note that.)and4.) see later in this work) are special cases of a more general classofnonlinearodesoftheform y where f y) is a rational function of y,namely, xx qx)y x + f y)y x = gx, y),.) f y) = Γ my m + Γ m y m + + Γ 0 y n + A n y n + + A 0, m,n Z +,.) qx)isanarbitraryfunctionofx, while y x and y xx denote the first and the second derivatives of the function yx) with respect to x, that is, dy/dx and d y/dx, respectively. Furthermore, A i, i = 0,...,n, and Γ j, j = 0,...,m, in.) are real constants and gx, y) in.) is a function possessing continuous partial derivatives with respect to x and y. In their work, Polyanin and Zaitsev [] present several equations subjected to the form.) and give their solutions or basic transformations and the resulted reductions of the

3 M. P. Markakis 3 Table.. Cases of the function f equation.)). f y) Equations in [].8..4) a = arbitrary constant ),.8..48),.9.3.3),.9.3.3),.9.4.).8..65), ) y a, a = arbitrary constant.6.4.6), ),.6.4.),.8..60),.9.3.0) y ay m, a = arbitrary constant, m.6.4.5),.6.4.7),.6.4.3) considered equations. In most of them, the function f y) has a rather simple form, as shown in Table. all the equations cited in this table refer to []). We also mention Langmuir s equation [, Chapter 7, Section, equation 9)] with f y) = /3)/y). In most of the above equations gx, y) is equal to 0, while in the others is a polynomial of y,thecoefficients of which are constants or functions of x. Moreover, qx) has a specific form in all equations except in [, equations ),.9.3.0),.9.4.)], where it is arbitrary. On the other hand, studying equations where f y) isan arbitrary function of y [, equations ),.9.4.3),.9.4.8)] with g = 0and [, equations.9.3.6),.9.3.3),.9.4.9)] with g an arbitrary function of y multiplied by e x in [, equation.9.3.6)]), we see that the proposed analytical methods result in integral equations, where even all the involved integrals can be determined, we may not derive a function y = yx). Furthermore, as far as.) Gambier s) as well as the 5th Painlevé transcendent [, equation.8..6)] are concerned both of the form.)),we have f y) = 3 y 4 y, qx)arbitrary, gx, y) = 0.3) y equation.)), f y) = 3y y y, qx) = x, gx, y) = y ) ay + β ).4) γ y)+δxyy +) x + y x y) 5th Painlevé transcendent). We also refer to [, equation.8..67)] with f y) = y y b, qx) = x a, gx, y) = 0..5) x In this paper we treat analytically the constrained case equation 4.)) f y) = Γ y + Γ 0 y, qx)arbitrary, gx, y) = 0,.6) b with b, Γ, Γ 0 arbitrary real parameters b>0, Γ + Γ 0 0).

4 4 International Journal of Mathematics and Mathematical Sciences. Existence of an analytical solution of an equation due to Gambier Developing the theory of second-order differential equations, Davis [, Chapter 7, Section 3] presents a solution for the following equation: y y ) y xx qx) y y ) y x 3 y )y x = 0, q arbitrary..) 4 The above equation is due to B. Gambier; and the solution is obtained by means of the transformation see []) where ξx) is a solution of the equation yx) = h [ ξx) ],.) ξ xx = qx)ξ x..3) Successive integrations of the latter equation furnishe ξx) = c e qx)dx dx + c,.4) with c, c being integration constants we can perfectly take the values and 0 for c and c,resp.).thus,bydifferentiating twice.) with respect to x, substituting the obtained expressions together with.)into.), and finally making use of.3), we manage to eliminatethe first derivativeand bring.) under the form h ξξ 3 4 h + ) h ξ = 0,.5) h where h ξ and h ξξ denote the first and second derivatives of hξ) with respect to ξ. Moreover, the specific treatment presented in [] introduces the substitution hξ) = [ Q ξ) ],.6) where Qξ) = Qξ, a, a 0 ) is the elliptic function of Weierstrass, which possesses the property see []) + u = Qξ) = ξ = Q ds u) =..7) u 4s3 + a s + a 0 By use of.7) we easily derive the following relations: Q ξ = 4Q 3 + a Q + a 0,.8a) Q ξξ = 6Q + a..8b)

5 M. P. Markakis 5 Then, by differentiating successively.6) with respect to ξ and making use of the obtained expressions for h ξ and h ξξ, together with.6),.5)isreducedto Q Q 3) Q ξξ + 3Q ) Q ξ = 0..9) Furthermore, introduction of.8) into the latter equation returns the polynomial form 4+a ) Q 3 +3a 0 Q a 0 = 0..0) Finally, by requiring that.0) holdstrueforeveryvalueofqξ), we extract the values a = 4, a 0 = 0. Thus, taking into account.),.4), and.6), we obtain the explicit solution of.), namely, yx) = Q ξ,4,0), ξx) = c e qx)dx dx + c..) 3. Generalization of the analytical treatment: development of aconstructivetechnique The idea is to set hξ) = ε P ξ)+ε Pξ)+ε 0, κ 0, 3.) instead of.6), where ε, ε, ε 0 are parameters and Pξ) is a function of ξ, the first derivative of which is assumed to satisfy the relation P ξ = a n P n + a n P n + + a 0, a i = constants, a n 0. 3.) The latter equation yields P ξξ = na n Pn + n )a n P n a. 3.3) Thus, evaluating h ξ and h ξξ by means of 3.), introducing the results together with 3.) into.5), and finally substituting 3.) and3.3), after some algebra, we obtain a polynomial of P in a more general form than that presented in.0), namely, 43 n)ε 3 a n P n n)ε ε a n P n+3 + =0. 3.4)

6 6 International Journal of Mathematics and Mathematical Sciences In addition, the requirement that 3.4) holds true for everyvalue of Pξ) inside an interval PJ), with J being the domain of ξx), results in a system of equations, formed by setting the coefficients of P m, m = n +4,n +3,...,0, equal to zero. Thus, by means of the P n+4 equation a n, ε 0) we determine n: For this specific value of n,3.)and3.3)become n = ) P ξ = a 3 P 3 + a P + a 0, a 3 0, 3.6a) P ξξ = 3 a 3P + a. 3.6b) Therefore, the function Pξ) = u can be defined as the inverse function of the generalized integral + ξ = P ds u) =, 3.7a) u a3 s 3 + a s + a 0 on condition that the integral converges, or can be defined as the inverse function of the indefinite integral ξ = P du u) =. 3.7b) a3 u 3 + a u + a 0 Note that Pξ) must be a continuousfunctioninside the domain of ξx). Furthermore, after substituting n = 3 into the set of equations furnished from the coefficients of the P polynomial 3.4), then by using the P 6 equation we obtain a 3, ε 0) Moreover, taking into account 3.9), the P 4 equation results in 6a 3 ε ε = 0, 3.8) ε = ) a 0 ε 3 = 0 3.0) a 0 = 0. 3.) Finally, after the obtained values ε = a 0 = 0 have been substituted into all the equations of the system n = 3), then the remaining equations corresponding to the odd powers of P become ε 0 a ε0 ) = 0, ) a ε +5a 3 ε 0 ε0 ) = 0, 3.) ) a ε + a 3 3 5ε0 = 0.

7 The system of the above equations yields the following four cases of solution. Case. Case. Case 3. Case 4. M. P. Markakis 7 ε 0 = a = a 3 = ) a = a 3 = ) ε 0 =, a 3 = ε a. 3.5) ε 0 =, a = a 3 = ) It is obvious that Cases,,and4 have to be rejected a 3 0) and thus the solution of the system 3.)is provided by Case 3 together with 3.9) and3.), namely, ε = a 0 = 0, ε 0 =, a 3 = ε a. 3.7) Therefore, the solution of.) isconstructedbymeansof.) and3.), where the parameters included in 3.)and3.6a)areasin3.7). We write yx)0 = ε P [ ξx) ] +, ξx) = e qx)dx dx 3.8a) P ξ = a P ε P + ), P ξξ = a 3ε P + ). 3.8b) Differentiating twice the above expression of yx), then introducing y, y x, y xx into.) and taking into account the equation giving ξx), that is, ξ xx qx)ξ x = 0, 3.9) as well as the expressions 3.8b), we see that.) is verified. Thus, the developed technique is able to construct an exact analytical solution of.). We note here that the solution 3.8a) has a general form yielding to.) as a special case, since if we take a 3 = 4 and assume that 3.7a) holdstrue,thenpξ) becomes the elliptic function of Weierstrass: Pξ) = Q ) ξ, a, a 0 = Q ξ, 4 ),0. 3.0) ε Moreover, by setting ε = weobtainsolution.).

8 8 International Journal of Mathematics and Mathematical Sciences 4. Use of the technique for another nonlinear ODE We apply now the technique developed above, in order to solve analytically the following equation: y b ) y xx qx) y b ) y x + ) Γ y + Γ 0 y x = 0, 4.) where qx) is an arbitrary function of x and b, Γ, Γ 0 are parameters taking real values, with b>0andγ + Γ 0 0. We observe that if in the nonlinear terms yy xx and yy x of.) we replace y with b and take the values 3/and3/4forΓ and Γ 0, respectively, then we obtain 4.). Moreover, 4.) is subjected to appropriate initial conditions,namely, y x 0 ) = y0, y x 0 ) = y 0. 4.) Thus, we simply follow the steps of the above procedure, as it was generalized in Section 3. More precisely we perform the following steps. ) We use the transformation Successive differentiations of 4.3)yield yx) = h [ ξx) ]. 4.3) y x = h ξ ξ x, y xx = h ξξ ξ x + h ξ ξ xx. 4.4) ) Similarly to the treatment of.) see Section ), we require that ξ xx = qx)ξ x, bymeansofwhichwecompute ξx) = c e qx)dx dx + c, 4.5a) 4.5b) with c, c being constants of integration. 3) By using 4.4) and4.5a), we reduce 4.)to h ξξ + Γ b h b + Γ ) h ξ = 0, 4.6) h + b with Γ = bγ + Γ 0 and Γ = bγ Γ 0. 4) We introduce a new function ξ) by setting hξ) = κ, κ 0, 4.7) ξ)+λ ξ)+μ with κ, λ, μ being real parameters, while we assume that ξ satisfies ξ = b n n + b n n + + b 0, b i = constants, b n )

9 M. P. Markakis 9 Differentiation of 4.8) with respect toξ yields ξξ = nb n n + n )b n n b. 4.9) 5) By using 4.7), 4.8), and 4.9), after some algebra, we finally transform 4.6)to apolynomialof,namely, ) 7 n 6)κ 4 b n n+6 + n 8 κ 3 λb n n+5 + =0. 4.0) 6) We form a system of equations by setting the coefficients of the polynomial 4.0) equal to zero. Thus, by means of the n+6 equation b n, κ 0) we determine n: For this specific value of n,4.8)and4.9)taketheform n = 6. 4.) ξ b b b + b + b 0, b a) ξξ = 3b 6 5 +b b 3 + b + b. 4.b) 7) We define the function ξ) = u, as the inverse function of the indefinite integral ξ = du u) = S =. b6 u 6 + b 4 u 4 + b 3 u 3 + b u + b u + b 0 4.3) ξ) should be a continuous function inside the interval J in which ξx)istaking values. Further, an investigation with respect to the determination of ξ) is presented below Section 5). 8) By making use of the set of equations established in step 6), we proceed to the evaluation of the constants and parameters involved in 4.7) and4.a). Thus, after substitution of n = 6 into all the equations of the system, the equation yields b 6, κ 0) Taking now into account 4.5), the 9 equation furnishes κ 0) 3b 6 κ 3 λ = 0 4.4) λ = ) 3b 3 κ 4 = 0 4.6) b 3 = )

10 0 International Journal of Mathematics and Mathematical Sciences Furthermore, after substituting 4.5) and4.7) into the equations of the system, then, by means of the resulted 7 equation we have κ 0) 5b κ 4 = 0, 4.8) b = ) Finally, after all the obtained results, that is, λ = b 3 = b = 0, have been substituted into the original system n = 6), then the remaining equations corresponding to the zeroth and even powers of take the following form: b 0 μ μ r ) = 0, b 0 κ [ Γ +3 ) r +Γ 0 r μ μ ] +b μ μ r ) = 0, b 0 κ Γ 0 r 5μ ) +b κ [ Γ + ) r + Γ 0 r μ + μ ] +3b 4 μ μ r ) = 0, 3b 0 κ 3 +b κ Γ 0 r μ ) + b 4 κ [ r Γ 0 μ + Γ ) + r +5μ ] +4b 6 μ μ r ) = 0, b κ + b 4 κ Γ 0 r + μ ) +b 6 [ r Γ 0 μ + Γ ) +4μ ] = 0, b 4 κ +b 6 Γ0 r +μ ) = 0, 4.0) where r = b. 4.) After performing algebraic manipulations, the system of six equations 4.0) results in the casesofsolutionlistedin Table 4.,where Γ 0 = Γ 0 b. 4.) We can see that these solutions refer to specific values of Γ 0 and Γ. We must also note here that κ and one of b 0, b, b 4, b 6 in most cases b 0 or b ) play the role of the free parameters involved in the final solution, the evaluation of which will be achieved by means of the initial conditions referring to 4.). 5. Determination of the function ξ) In step 7) of the analytical procedure developed in Section 4, we have considered the function ξ) = u as the inverse function of the indefinite integral S ξ = Su), expression 4.3)). Thus, after the determination of S by elementary or special functions) we should solve explicitly or implicitly the obtained relation with respect to u, thus determining the function ξ), that is, ξ = u) = Su) = u = ξ). 5.)

11 μ 0 0 ± b ± b ± b ± b 0 ) Γ0,Γ ±, 3 ) ± 3, 3 ) 0, ) ±, ) ±, 3 ), ) ±, ) Table 4.. Solutions of the system 4.0). M. P. Markakis b 6 b 4 b b 0 b 3 κ 3 b 0 b κ b 0 ±bκb 0 ±b 3 κ 3 b 0 3b κ b 0 ±3bκb 0 ± b3 κ3 b 0 b κ b 0 ± 5b κb 0 ± b3 8 κ3 b 0 3b 4 κ b 0 ± 3b κb 0 ± b3 4 κ3 b 0 5b 4 κ b 0 ±bκb 0 ±b 3 κ 3 b 0 3b κ b 0 ±3bκb 0 ±bκb , ) b κ b ±, ) b κ b ±bκb 0 ± b ± b ± b ± b ± b ± b, ) 0 0 0, ) ± b κb , 3 ) ±bκb b 0, ) 4 κ b ±bκb 0, ) b κ b ±bκb 0, 3 ) b κ b ± 3b κb 0 If an explicit expression unique or not) uξ) cannot be obtained, then it should be investigated if a function uξ) can be determined implicitly by means of the equation ξ Su) = 0. This means that we should examine the well-known conditions of the implicit function theorem for this we need to locate a point ξ 0,u 0 )suchthatξ 0 = Su 0 )and then study the existence and continuity of the derivative S u) inside an interval containing u 0 and evaluate S u 0 )). If the conditions of the theorem are satisfied, then a unique implicit smooth function u = ξ) isdefinedinsideacertainareaofξ 0,u 0 ). Note that, in case of a singular point, a nonunique function may be defined. In any case of solution of 5.), the formula 4.7)withλ = 0) combined with 4.5b)withc =, c = 0) as well as with the results given by Table 4. represents an exact solution of 4.) inside a certain domain of x, where the free parameters are evaluatedby the use of initial conditions 4.). Technically, by making use of the substitution see [3, equation.9.)]) u = z, 5.)

12 International Journal of Mathematics and Mathematical Sciences the integral S 4.3), b = b 3 = 0), du S =, 5.3) b6 u 6 + b 4 u 4 + b u + b 0 is transformed to S = dz z ), u>0, 5.4) b 6 z 3 + b 4 z + b z + b 0 wherewehaveconsideredu to take positive values in the opposite case u<0), a sign appears in the right-hand side of 5.4)). Moreover, the integral 5.4) isdetermined in dependence on the case of solution Table 4.). Thus, as far as the cases with b 0 = 0are concerned, S becomes S = S 0 = dz z R z), R z) = b 6 z + b 4 z + b, 5.5) the various types of which depending on the sign of b or b 6 ) and the discriminant Δ of the quadratic R )areprovidedby[3, equation.66)], while for the cases where b 0 0, the integral S takes the form S = S = dξ ±b 6 ±zr3 z), b 6 0, 5.6) with R 3 z) = z 3 + c 4 z + c z + c 0, 5.7a) c i = b i, i = 0,,4. 5.7b) b 6 The sign ± in5.6) corresponds to the positive and negative signs of b 6, respectively. By using now the substitution the cubic form R 3 can be written as z = ω c 4 3, 5.8) R 3 ω) = ω 3 + pω + q, 5.9) with p = c c, q = 7 c3 4 c 4c 3 + c ) As it is well-known [4, Cardan solution], the roots of R 3 ω) depend on the sign of the discriminant Δ 3 = p3 7 + q 4. 5.)

13 M. P. Markakis 3 Furthermore, by introducing 5.7b)and5.0)into5.), we find that Δ 3 is equal to zero in all casesofsolutionrecorded in Table 4.. Therefore, according to the Cardan solution, R 3 z) has three real roots two of them are equal), provided by ρ i = e i c 4 3, i =,, e = 3 q, e = 3 q, 5.) where the relation 5.8) has been taken into account. Here ρ is the root of double multiplicity. Thus, the integralsformula 5.6)) becomes S = dz ) ±b 6 z ρ ±z ρ z. 5.3) By introducing now the substitution see [3, equation.8)] with n = ) t =, z ρ z>ρ, 5.4) the integral 5.3)istransformedto S = dt ±b 6 γt + βt + a, 5.5) with ) γ = ρ ρ ρ, β = ) ρ ρ, a =±. 5.6) Finally, the latter integral can be computed by means of [3, equation.6)] the formulas by means of which the integral 5.5) is expressed depend on the sign of γ and the discriminant Δ of the quadratic γt + βt + a). It must be noted here that all the expressions concerning both integrals, ξ = S 0 u) formula5.5)) and ξ = S u) formula 5.5)), can be solved explicitly with respect to z = u. Thus, the function u = ξ) can be determined by taking the positive square root of the obtained expressions, since we have considered u to be positive see the integral 5.4)above). A question emerging here concerns the form of the integral S when Δ 3 0. Obviously, this is not the case of 4.) but it is quite possible to occur in this step of the followed procedure in case that it is applied to another nonlinear equation. Thus, when the sign of the discriminant Δ 3 is positive R 3 z) has one real and two conjugate complex roots) or negative R 3 z) has three distinct real roots), then by developing the expression ±R 3 z) and taking into account the integral 5.6), we deduce that only generalized integrals of thesecondkindoftheform b a dt tr3 t) 5.7)

14 4 International Journal of Mathematics and Mathematical Sciences can be determined by explicit formulas. In the above integral either, a or b) represents z = u and b or a) is equal to one of the real roots of R 3. More precisely, the integral 5.7) is expressed by means of elliptic integrals of the first kind F [ f z, p,q,c 4 ),κ p,q,c4 )], 5.8) where c 4 and p, q areasin5.7b) and5.0), respectively. As it is well known, F can be solved with respect to z by means of the Jacobi elliptic sine function. Therefore, in order to finally obtain an integral of the kind 5.7), the integral S shouldbedefinedasa definite integral of the form b0 ds S =, 5.9) a 0 b6 s 6 + b 4 s 4 + b s + b 0 with either a 0 or b 0 ) being equal to u and b 0 or a 0 ) being equal to the square root of one of the real roots of R Application of the method to special cases of 4.) We now consider two special cases of 4.), for which the parameters of the solution are provided by Table 4..More precisely,we apply the analysisdeveloped in Section 5 to the cases i) Γ 0 = 0, Γ = : ii) Γ 0 = 3/)b, Γ = 3/: y b ) y xx qx) y b ) y x yy x = 0, 6.) y b ) y xx qx) y b ) y x 3 y b)y x = 0. 6.) Equation 6.). Here, twopossible solutionsare recorded intable 4.. Case. We consider the + case) μ = b, b 6 = b3 κ3 b 0, b 4 = b κ b 0, b = 5b κb ) Since b 0 0, by making use of the above relations and combining 5.7b)with5.0), we obtain the roots of R 3 z)δ 3 = 0), namely, ρ = bκ, ρ = bκ, ρ is double. 6.4) Then, by substituting 6.4) into5.6), the integral S S = S )givenbyformula5.5) becomes ξ = S = dt ±b 6 ρt ±, t = > ) z ρ

15 M. P. Markakis 5 The signs in 6.5) correspond to the cases b 6 > 0andb 6 < 0, respectively. According to [3, equation.6)], S is determined as ξ = arcsin ρ, b 6 > 0 6.6a) b 6 ρ z ρ ξ = ln ρ z ) z ρ +ρ, b 6 < 0, ρ > b) b 6 ρ z ρ Moreover, solution of 6.6) with respect to z = u, respectively, results in u = ρ + ρ sin b 6 ρ ξ ), b 6 > 0, 6.7a) u = ρ + 4ρ e b 6ρ ξ e 4, b b 6ρ ξ +4ρ 6 < b) Since ξ) = u can be determined as the positive square root of the right-hand side of 6.7)), the relations 4.3)and4.7)λ = 0, μ = /b)yield yx) = bsin [ b 6 ρ ξx) ], b 6 > 0, 6.8a) yx) = b e 4 b 6ρ ξx) +4ρ 4ρ e, b 6 < 0, 6.8b) b 6ρ ξx) where the expression of ρ equation 6.4)) has been taken into account. Differentiating twice 6.8) with respect to x, introducing the obtained expressions into 6.), and taking into account 4.5a), we see that 6.) is satisfied in both cases. Therefore, 6.8) represent an exact explicit solution of 6.), where b 6 and ρ stand for the free parameters instead of κ, b 0 see relations 6.3), 6.4)). We now examine the second case. Case. μ = 0, b 6 = b κ b, b 4 = 0, b 0 = ) Since in this case we have that b 0 = 0, the corresponding integral S S = S 0 )givennowby 5.5)becomes ξ = S 0 = dz z. 6.0) b 6 z + b According to [3, equation.66)], we havethe followingtwo cases: ξ = b arcsin bκz, b < 0, 6.a)

16 6 International Journal of Mathematics and Mathematical Sciences where the signs correspond to κ>0andκ<0, respectively b 6 has been substituted from 6.9)), and ξ = ln b b6 z ) + b +b, b > 0. 6.b) b z Furthermore, by solving 6.) with respect to z = u and substituting b 6 from 6.9), we obtain u = b κ sin b ξ ), b < 0, 6.a) u 4b e b ξ = e 4, b b ξ +4b κ b > 0. 6.b) Thus, substituting the above expressions of ξ) = u into 4.7) λ = μ = 0), then by 4.3), we extract yx) = bsin [ b ξx) ], b < 0, 6.3a) yx) = e 4 b ξx) +4b κ b 4κb e, b > b) b ξx) The sign in6.3a)holdstruewhenκ>0, while the sign + holds true when κ<0. Moreover, introducing y as well as its derivatives with respect to x) into6.) and taking into account 4.5a), we deduce that 6.) is also verified in both cases. Comparing with the expressions 6.8) obtainedincase, we see that the explicit forms of the obtained solutions are qualitatively the same. We mention that ξx) isgivenfrom4.5b), namely, ξx) = c e qx)dx dx + c 6.4) we can take c =, c = 0). Moreover, the respective to the case under consideration, free parameters b 6, ρ or κ, b ) can be evaluated by means of the initial conditions referring to original equation 6.) relations4.)). These conditions will also be responsible for thekindofsolution 6.3a) or6.3b), 6.8a) or6.8b)) that 6.) will accept. It is worth to be noted here that solution 6.3a) or6.8a)) is in perfect agreement with the solution of the same equation with qx) = x/a x ) given by Polyanin and Zaitsev [, equation.8..67)], namely, yx) = bsin A arcsin x ) a + A, A, A = constants. 6.5) In fact, for this specific form of qx), 6.4)yields ξx) = c arcsin x a + c, 6.6)

17 M. P. Markakis 7 and therefore 6.3a)becomes yx) = bsin b c arcsin x ) b a + c, 6.7) or setting B i = b c i, i =,, yx) = bsin B arcsin x ) a + B, 6.8) which is identical to 6.5). Equation 6.). Table 4. also provides two possible solutions concerning this equation, namely, the following. Case. μ = 0, b 6 = b 3 κ 3 b 0, b 4 = 3b κ b 0, b = 3bκb ) This case b 0 0), as the case i) of 6.), corresponds to the integral S S = S )expression 5.6)), involving the cubic form R 3 ξ). By means now of 5.7b)and5.0), the roots of R 3 Δ 3 = 0) are found: ρ = ρ = bκ, ρ is double, 6.0) and moreover combining 6.0)with5.6), the obtained above integral 5.5) takes the form ξ = S = dt ±b 6 ±ρ t ±, t = > 0, z ρ = ± z, ρ ±b6 z ρ 6.) where the signs correspond to the positive and negative signs of b 6, respectively. Since z/z ρ ) is positive, we reject the case b 6 < 0, writing ξ = z, 6.) ρ b6 z ρ and therefore, z = u b 0 ξ = bκb 0 ξ, κb 0 > 0, 6.3) where b 6 and ρ have been substituted from 6.9)and6.0). Thus proceeding as in the previous case, we derive yx) = b + κb 0 ξ x), κb 0 > )

18 8 International Journal of Mathematics and Mathematical Sciences Moreover, 6.) is satisfied by formula 6.4), where κ and b 0 represent here the free parameters. Case. μ = b, b 6 = bκb 4, b = b 0 = ) Here, the integral S is given by formula 5.5)b 0 = 0).Moreprecisely,wewrite ξ = S 0 = Therefore, according to [3, equation.66)], we have and moreover, dz z b 6 z + b 4 z. 6.6) b6 z ξ = + b 4 z, 6.7) b 4 z z = u = b 4 ξ + bκ, 6.8) where b 6 has been substituted from 6.5). Then, similarly as before, yx)isobtained: yx) = b κb b 4 ξ x). 6.9) The latter formula also verifies 6.). As in 6.), the function ξx) isprovidedby6.4) c =, c = 0), while the free parameters κ, b 0 or κ, b 4 ) corresponding to Cases and are evaluated by the associated initial conditions. Finally, we also observe here the qualitative similarity between the expressions giving yx)incases and. 7. The functions x) According to the developed technique, the determination of the function is achieved through the inversion of the integral 4.3), which in turn can be determined by means of classical procedures Section 5). Considering now the functions x) involvedinthe obtained exact solutions of 6.)and6.)withξx)asindependentvariable),provided by the expressions 6.), 6.7)and6.8), 6.3), respectively we take the positive square roots of the right-hand sides), we observe that their form is simply a specific combination of transcendental or and) classical functions. On the other hand, we can see that they share the fundamental property of elliptic functions, that their only movable singularities are poles. More specifically, taking arbitrary values for the respective free parameters, we have b>0).

19 M. P. Markakis 9 x) x a) The function b =, κ =±, b = ) x) x b) The function b =, ρ =, b 6 = ) 6 Figure 7. Equation 6.). Formula 6.a), b < 0: Function Figure 7.a)) [ x) = b κ csc b x )] /, 7.) where x λ +)/ b )π,λ +)/ b )π), λ Z. possesses obvious poles at x = λπ b, λ Z. 7.)

20 0 International Journal of Mathematics and Mathematical Sciences Formula 6.7a), b 6 > 0: Function [ x) = ρ csc b 6 ρ x ))] /, ρ = bκ, 7.3) where x λ +)/ b 6 ρ )π,λ +)/ b 6 ρ )π), λ Z. possesses obvious poles at x = λπ b 6 ρ, λ Z. 7.4) For ρ > 0, presents a qualitatively similar graph to that of in Figure 7.a) displaced upwards), while for ρ < 0weobtainFigure 7.b). We observe that in this case, is not a smooth function inside its domain, since it accepts different values for its right-side and left-side derivatives at x 0 = λ +)/ b 6 ρ )π + π/4 b 6 ρ, λ Z, equal to b 0 and b 0, respectively in Figure 7.b), b 0 = ). This result can also be derived from 4.a)bysetting x 0 ) = x 0 ) = 0. Formula 6.b), b > 0: Function Figures 7.a), 7.b)), x) = b e b x e 4 b x +4b κ b ) /, x R. 7.5) possesses no poles. The function corresponding to formula 6.7b) defined everywhere in R, possessing no poles, and valid only for ρ > 0) presents also qualitatively similar graphs to that of, displaced upwards and stretched or compressed along x and y axes, depending on the values of parameters b 6, ρ. Equation 6.). Formula 6.8): Function Figures 7.3, 7.4, 7.5) ) / x) = b 4 x. 7.6) + bκ We have the following three cases: i) b 4 > 0 κ>0: x) = x), x R which possesses no poles) 7.7) ii) b 4 > 0, κ<0: x) = x), ) x, bκ/b 4 bκ/b 4,+ ), 7.8) iii) b 4 < 0, κ>0: x) = 3 x), ) x bκ/b 4, bκ/b )

21 M. P. Markakis x) x a) The function b =, κ =±, b = ) 6 x) x b) The function b =, κ =±0, b = ) Figure 7. Both and 3 possess two poles at x = bκ b 4, bκ b ) Finally, in the case b 4 < 0, κ<0, does not accept real values for any x R. Formula 6.3): Function b x) 0 x ) / = bκb 0 x, κb 0 > 0. 7.)

22 International Journal of Mathematics and Mathematical Sciences x) x 5 0 Figure 7.3. The function b =, κ =, b 4 = ). x) x 5 0 Figure 7.4. The function b =, κ =, b 4 = ). We have the following two cases: i) b 0 < 0: x) = x), ) ) x,,, 7.) bκb 0 bκb 0 ii) b 0 > 0: x) = 3 x), ) x,. 7.3) bκb 0 bκb 0

23 M. P. Markakis 3 3 x) x 3 Figure 7.5. The function 3 b =, κ =, b 4 = ). 3 x) x 3 Figure 7.6. The function 3 b =, κ =, b 0 = ). Both and 3 possess two poles at x =,. 7.4) bκb 0 bκb 0 The graph of is similar to that of in Figure 7.4, while the respective graph to 3 is presented in Figure 7.6.Weseethat 3,as ρ < 0), is also a nonsmooth function since the right-side and the left-side derivatives at x = 0takedifferent values, equal to b 0 and b 0, respectively.

24 4 International Journal of Mathematics and Mathematical Sciences 8. Summary: conclusion By the developed analytical procedure, we first obtained a more general form of the solution of Gambier s equation.) than the existed explicit one. We then applied this method to another nonlinear ODE equation 4.)) extracting specific sets of values of the parameters involved, for which a closed form solution can be obtained. This solution is constructed through the inverse function of the integral S of the algebraic) inverse of the square root of a polynomial of the 6th degree, the specific form of which allows the determination of S Section 5). Thus, by inverting the extracted relations we determined explicitly or implicitly the function introduced into the analysis and derived the respesctive explicit or implicit) formula of the solution. Two arbitrary parameters contained in the obtained expressions provide special solutions in case of an initial value problem. Furthermore, we note that in the special cases of 4.) solved in this work as an application of the developed technique Section 6), the corresponding functions ij and ij Section 7) as well as the corresponding explicit) formulas of the function y Section 6) represent specific combinations of elementary functions. Moreover, the most of the obtained functions possess movable poles. We should also note the perfect agreement between the obtained solution and that given in [] in case of6.). More generally speaking, we believe that the establishment of analytical techniques in the sense of constructing successive appropriate steps, as in the present work, can achieve the derivation of analytical solutions for several nonsolved analytically up today nonlinear ODEs or systems of ODEs. References [] H.T.Davis,Introduction to Nonlinear Differential and Integral Equations,Dover,NewYork,NY, USA, 96. [] E. Picard, Traité d analyse, vol., Gauthier-Villars, Paris, France, 893. [3] E. Picard, Traité d analyse, vol. 3, Gauthier-Villars, Paris, France, 896. [4] E. Picard, Sur un classe d equations differentielles, Comptes Rendus, vol. 04, pp. 4 43, 887. [5] P. Painlevé, Sur la determination explicite des equations differentielles du second ordre à points critiques pixes, Comptes Rendus, vol. 6, pp , 898. [6] P. Painlevé, Sur les equations differentielles du second ordre à points critiques fixes, Comptes Rendus, vol. 6, pp , , 898. [7] P. Painlevé, Sur les equations differentielles du second ordre à points critiques fixes, Comptes Rendus, vol. 7, pp , , 898. [8] P. Painlevé, Sur les equations differentielles du second ordre à points critiques fixes, Comptes Rendus, vol. 9, pp , , 899. [9] B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l intégrale générale est a points critiques fixes, Acta Mathematica, vol. 33, no., pp. 55, 90. [0] E. L. Ince, Ordinary DifferentialEquations, Dover, New York, NY, USA, 97. [] M. P. Markakis, On the reduction of non-linear oscillator-equations to Abel forms, Applied Mathematics and Computation, vol. 57, no., pp , 004. [] A.D.PolyaninandV.F.Zaitsev,Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, nd edition, 003.

25 M. P. Markakis 5 [3] I. J. Grabshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, San Diego, Calif, USA, 5th edition, 994. [4] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, NY, USA, nd edition, 968. M. P. Markakis: Department of Engineering Sciences, University of Patras, 6500 Patras, Greece address: m.markakis@des.upatras.gr