Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1

Abstract The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations yy′=(αxm+k−1+βxm−k−1)y+γx2m−2k−1,y′=dydx $$\begin{array}{} \displaystyle yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}, \quad y'=\frac{dy}{dx} \end{array}$$ where a, b, c ∈ ℂ, m, k ∈ ℤ and α=a(2m+k)β=b(2m−k),γ=−(a2mx4k+cx2k+b2m). $$\begin{array}{} \displaystyle \alpha=a(2m+k) \quad \beta=b(2m-k), \quad \gamma=-(a^2mx^{4k}+cx^{2k}+b^2m). \end{array}$$ This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercise presented a typo which does not allow to solve correctly it. We present the corrected exercise, which corresponds to the title of this paper. We solve the exercise and afterwards we make algebraic and of singularities studies to this family of differential equations.


Introduction
Dynamical systems is a topic of interest for a large number of theoretical physicist and mathematicians due to the seminal works of H. Poincaré. It is well known that any dynamical system is a system which evolves in the time. H. Poincaré introduced the qualitative approach to study dynamical systems, which has been useful to study theoretical aspects and applications to biology, chemistry, physics, among others, see [1][2][3][4][5][6][7].
On another hand, E. Picard and E. Vessiot introduced an algebraic approach to study linear di erential equations based on the Galois theory for polynomials, see [8][9][10][11][12][13]. Combination of dynamical systems with di erential Galois theory is a recent topic which started with the works of J.J Morales-Ruiz (see [12] and references therein) and with the works of J.-A. Weil (see [14]). Further works about applications of di erential Galois theory include [15][16][17].
The Handbook of Exact Solutions of Ordinary Di erential Equations, see [18], is one important reference for scientists and engineers interested in solving explicitly ordinary di erential equations. This book contains around 3,000 nonlinear ordinary di erential equations with solutions, as well as exact, symbolic, and numerical methods for solving nonlinear equations. Nonlinear equations and systems with rst-, second-, third-, fourth-, and higher-order are considered there.
Inspired by a previous version of the paper [19], we analysed the Exercise 11 in [18, §1.3.3], which corresponds to a ve parametric family of di erential equations. We discovered a typo (also corrected by us), which was corrected in the nal version of [19] to study from di erential Galois Theory point of view the integrability of the dynamical system proposed in such excercise.
We call as Polyanin-Zaitsev vector eld to the vector eld associated to this system of di erential equations that comes from the corrigendum of the Exercise 11 in [18, §1.3.3]. Moreover, we study integrability aspects using di erential Galois theory, following [9,19] as well qualitative aspects due to the foliation associated to Polyanin-Zaitzev vector eld is a Liénard equation, which is closely related to a Van Der Pol equation.
This paper not only present the corrigendum and complete solution of the Polyanin-Zaitsev excercise mentioned above, it also extends the results given in [19] concerning the Polyanin-Zaitsev vector eld. From algebraic point of view we give conditions over the parameters to have polynomial vector eld, moreover we obtain the critical points for some particular cases and we describe their behavior.
The results of this paper were obtained, but not published, during the seminar Algebraic Methods in Dynamical Systems in 2013 developed by the rst author and in the master thesis of the second author in 2014 (supervised by the rst and third author).

Preliminaries
In this section we provide the necessary theoretical background to understand the rest of the paper. A planar polynomial system of degree n is given bẏ being P, Q ∈ C[x, y] and n = max(deg P, deg Q). By X := (P, Q) we denote the polynomial vector eld associated to the system (1). The planar polynomial vector eld X can be also writen in the form A foliation of a polynomial vector eld of the form (1) is given by Following [6], we present the following theorem, which allow us the characterization of the critical points.
Then the following statements hold: a) If α is even and α > β + , then ( , ) is a saddle node. If α is even and α < β + or Φ(x) ≡ , then the ow near of ( , ) have two hyperbolic sectors. b) If α is odd and a > , then ( , ) is a saddle point. c) If α is odd and a < , several cases can occur: in this case the ow near to the critical point ( , ) is topologically conformed by an elliptic sector joint with an hyperbolic sector.
in this case the critical point ( , ) is focus or center.

Corrigendum to the problem
The original Exercise 11, section 1.3.3 of the book of Polyanin-Zaitsev (see [18, §1.3.3.11]) was presented as follows: The transformation z = x k , y = x m (t + ax k + bx −k ) leads to a Riccati equation with respect to z = z(t): The substitution z = mt +c ak w t w , where c = c − abm, reduces equation (2) to a second order linear equation: The transformation ξ = where ν is a root of the quadratic equation ν + ν + m −k m − abk mc = . A typo in this exercise does not allows its solving. The correction of the problem is presented in the following proposition: Proof. The system of equations, associated with this Liénard equation is: and di erentiating we have that: then, the associated foliation has the form ydy x m− k− as the Liénard equations. Now we compute each part of this equality, thus we obtain the left side as: and the right side as For our purpose, we organize the terms with respect to dz, that is: Now organizing the terms again we have: Thus, we obtain the Riccati equation: and we conclude the proof.
For the rest of transformations proposed in the Exercise of Polyanin-Zaitsev we need the results concerning the transformations, which will be given in the next section.

Some transformations
In this section we study some transformations that allow us to complete the exercise stated by Polyanin-Zaitsev above.
Then the di erential equation Proof. R ∂ x y + SR∂x y + Cy = then replacing We divide all by a , thus we get: Then: Replacing x in the polynomial S in term of τ, we get: Now, if λ = C a , the di erential equations will be Q ∂ τŷ + LQ∂τŷ + λŷ = , whereŷ = y(x(τ)), and the transformation τ = x + a a send ∂x on ∂τ.
Remark 3.1. The di erential equation of the form with λ = n(n + +ã +b) and n ∈ N, is known as Jacobi equation (in general form). It is a particular case of the hypergeometric equation, but the solutions include Jacobi polynomials. If we takeã =b andλ = n(n + ã) with n ∈ N, we get a Gegenbauer equation (or ultraspherical case): Now we study a special transformation, in the following theorem: The di erential equation an z (n) + (k + a n− )z (n− ) + a n− z (n− ) + ... + a z ( ) + a z = , with a i ∈ C(x), with an ≠ , can be transformed into the di erential equation Proof. Following the Lemma 3.1 and taking the implicit we compute the rst equation applying the change of variable Then, computing in general form the Leibniz rule we get: Continuing of this form, we divide all equation by ε. Then, we use the same method of the indeterminate coe cients to calculate ε and the b i coe cient: If we take y (n− ) : From this di erential equation we get an appropriate ε value, and with it we obtain the coe cient b i .
If we take y (n− ) : If we take y (n− ) : Continuing of this form we see, that for any k ∈ N the recurrent formulae will be: ε + a n−k n−k n−k = b n−k If in the previous theorem, we considerk = , we get the next corollary: Corollary 3.3. The di erential equation an z (n) + a n− z (n− ) + a n− z (n− ) + ... + a z ( ) + a z = with a i ∈ C(x) can be transformed in the following equation Example: Applying the transformation over the general second order di erential equation z +a z +a z = : Using Theorem 3.2, we get ε ε + a = , then ε = − a ε. Now, through derivatives and dividing by ε, we get ε = − a − ε ε a , but ε ε = − a . Thus, we obtain In this way we arrive to b = − a + a , for the di erential equation y + by = .
In the following theorem we recall that a Hamiltonian Change of Variable z = z(x) is a change of variable, where (z(x), z (x)) is a solution curve of a Hamiltonian system of one degree of freedom. The new derivation is given by∂z = √ α∂z, being α = (∂x z) , see [15][16][17] and references therein.
is transformed in the equation Owing to λ = n(n + +ã +b), we have the Jacobi equation withã =b. Moreover, if λ = n(n + ã) then we obtain a Gegenbauer equation.
Proof. Case 1: If we assume a = b = , we obtain l = , l = b a , q = a a . Then: Furthermore, due to ξ = ∂τ √ Q then α we arrive to: We compute all elements of the Hamiltonian change of variableQ w(ξ (τ)) = w(τ). Then: If we replace, in the transformed di erential equation, we obtain: It is equivalent to Gegenbauer equation withλ = λ q , − ã + = l − q i.eã = q −l − then: Finally if λ q = n(n + q − l − ) with n ∈ N, then the solutions of this equation are Ultraspherical Gegenbauer polynomial.

Case 2:
If b = a b a then: i.e the initial di erential equation will be: (τ + q ) ∂ τŵ +l τ(τ + q )∂τŵ + λŵ = for instance, the same di erential equation of the previous case.

Lemma 3.5. The Gegenbauer equation
Proof. For this transformation we will use a Hamiltonian change of variable, over the independent variable of the Gegenbauer equation x = − z. Then √ α = ∂x z = − , that is, α = ,∂z = ∂z and∂ z = ∂ z . Substituting in the Gegenbauer equation, we obtain Thus, we obtain the equation We know that Hypergeometric equation is of the form Then, we compute the parameters values a, b and c as follows: which concludes the proof.
Proof. Firstly we transform the Legendre equation into the Hypergeometric equation: Now dividing by (x − ) µ − and replacing in each terms of the Legendre equation we get: Now we obtain Now applying the Hamiltonian change of variable x = − ξ ξ = −x , we obtain ∂x ξ = − = √ α, being α = . Thus, we obtain∂ ξ = − ∂ ξ , then x − = − ξ + ξ − = ξ (ξ − ). Now, replacing we obtain: Applying the previous lemma we have the equation: Now applying the previous theorem we get: Remark 3.2. If we have our equation in the Legendre form, we apply the proposition 3.6 and therefore we can study it as in [19] to conclude the integrability or non-integrability, of the Liénard equation. Moreover, such as we will see in the next section, through equation (5) in Legendre's form we can apply the Kimura table, see [19].

Polyanin-Zaitsev vector eld
The associated system of the Polyanin-Zaitsev vector eld, with a, b, c, m, k ∈ R, is given by: with α = a( m + k), β = b( m − k) and γ = (a mx k + cx k + b m), where the Polyanin-Zaitsev vector eld is given by X =: (P, Q), being, The next proposition can illustrate the cases in which the Polyanin-Saitsev vector eld is formed by non trivial polynomial functions.
Proposition 4.1. The system (5) is a not null di erential polynomial system if it is equivalently to one of the next families:ẋ x = ẏ y = +b(m + p + )yx p − b mx p+ (11) x = ẏ y = a(m + s + )yx s − amx s+ (12) with s, p, r ∈ Z de ned in the proof.
Proof. The system (5) is a polynomial system if Q is a polynomial function, that is, the exponents of each term must be non negative integer. Furthermore, we need to consider the values of the constants a, b and c. Now we consider the di erent possibilities for these constants: Case 1. For a ≠ , b ≠ , c ≠ , it must be satis ed: being s, p, r ∈ Z + . Thus m = r+ and k = s−p , which means that r = s + p + . Therefore we obtain the following system associated with the family (5): x = ẏ y = [a s+p+ x s + b s+ p+ x p ]y − a s+p+ x s+ − cx s+p+ − b s+p+ x p+ .
Case 2. For a = , b ≠ , c ≠ , the system (5) is reduced to: since the exponents must be non-negative integers, then: for instance, we arrive to the system: Case 3. For a ≠ , b ≠ , c = , the system (5) is reduced to: again the exponents must be non-negative integers, therefore: thus, m = s+p+ and k = s−p , which lead us to the following system: Case 4. For a = , b = , c ≠ , the system (5) is reduced to: due to the exponents must be non-negative integers, we arrive to m − = r ∈ N, that is, m = r+ .
Case 5. For a ≠ , b = , c ≠ , the system (5) is reduced to: then m = r+ and k = s−r+ . Thus we arrive to the system: x = ẏ y = a s+p+ yx s − a r+ x s+ − cx r . Case 6. For a = , b ≠ , c = , the system (5) is reduced to: then there is a line of solutions, with r ∈ Z + . In this case the associated family is: Case 7. For a ≠ , b = , c = , the system (5) is reduced to: in this case: then the associated family is:ẋ = ẏ y = a(m + s + )yx s − amx s+ .

Finite critical points
In this section, we present an study about the existence of nite critical points and the stability for each family associated to the Polyanin-Zaitsev vector eld. Proof. For this proof we take the family (6) as form (5). Then, we will have to solve the system: If y = , then (a mx k + cx k + b m)x m− k− = , with m − k − ≥ , then for the product will be equal to , must be x = or (a mx k + cx k + b m) = . In the rst case we obtain that x = and we conclude (x, y) = ( , ).
For the second case, if we completing squares in the polynomial, we will have that: Now for the family (10), we have that y = a s+p+ yx s − a r+ x s+ − cx r = .
Then deg(Q) = max{r, s + }, again we have to consider two cases.
-If deg(Q) = s + then x r a r+ x s+ −r + c = . This implies that If γ is even then it is necessary that c < , and for instance the critical points are ( , ), γ − c a(r+ ) , and − γ − c a(r+ ) , . Proof.

Proof a. & b:
Over the conditions of (1.1): ( , ) is an isolated critical point.
The degree of Y(x, y) should be greater than .
Now checking the conditions of the theorem we have: α is odd,ā < ,b + ā(β + ) = a ( m + k) + a m(m + k) > , we have the conditions of item c) (1.1). If β is even andb > , then ( , ) is an unstable node. On the other hand, if β is odd, then the critical point is the union of an elliptical sector and with an hyperbolic sector.

Proof c.
If m = and k = / , c = − abm, c < , a > , b > in this case, taking d = k −c a m and the variable change u = x − d the behavior in a neighborhood of the critical points (± k −c a m , ), is topologically equivalent to behavior of the systemv in a neighborhood of the origin. The Jacobian matrix of (13) on the critical points ( , ) have the form Then the eigenvalues are: λ , = β − α = a d + abd + b d + a d + cd + b d . Now we will de ne the function Taking into account the condition c = − abm that is c = − ab, with a > , b > the resultant function is Now we going to found the critical points of this function. that is the roots of the equations H (x) = . As x + b , then the real roots are

An example:
Now we show the qualitative study of a particular case associate to the family (7): The critical points of this systems are: ( , ) and ( − b c , ). Now we studies the behavior the orbits near them.
For the vector eld X(x, y) = (y, by − b x − cx ) the linear part is: and his characteristic polynomials is: λ − bλ − b = , then the eigenvalues are λ = b and λ = − b. It is, the system has a saddle at the point. Now we will analysis the in nity behavior, using the Poincaré compacti cation. For this we will use the equivalent systems at the the chart U and U give by the variable change respectively. For more see [20].
At U chart the system is:u With critical point ( , − c b ), but this is not on the equator of the Poincaré sphere.
At U chart the system is:u

Final Remarks
In this paper we studied from algebraic point of view and singular points study the ve parametric family of linear di erential systems that came from the corrigendum of Exercise 11 in [18, §1.3.3], which we called Polyanin-Zaitsev vector eld. We solved the corrected exercise through a series of transformations using Hamiltonian changes of variables. A analysis was also developed to nd critical points and their behavior.