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On the second order homogeneous quadratic differential equation

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Let the elementsa, b, c, d, e, f belong to a differential field\(\mathfrak{F}\) of characteristic zero. This paper is concerned with the solutions belonging to\(\mathfrak{F}\) of the second order homogeneous quadratic differential equation

$$Q(y) \equiv ay''y'' + by''y' + cy''y + dy'y' + ey'y + fyy = 0$$
((i))

. IfQ(y) were multiplicatively reducible, then the problem of solving equation (i) becomes the easier problem of solving at worst second order homogeneous linear differential equations. We make the following assumptions:

(A)Q(y) is multiplicatively irreducible;y 2+1=0 has a solution in\(\mathfrak{F}\).

(B) Whenm 1 andm 0 are elements of\(\mathfrak{F}\) withm 1≠0, the differential equationm 1 y′+m 0 y=0 possesses a non-zero solution in\(\mathfrak{F}\).

(C) Elementsp, q, r, s of\(\mathfrak{F}\) exist so that equation (i) may be written in the form

$$Q(y) \equiv ay''y'' + by''y' + cy''y + (py' + qy)(ry' + 8y) = 0$$
((ii))

. Here, the naming of the elementsp, q andr, s could be interchanged. We agree, whenever possible, to choose a naming not giving validity to both of the conditions

$$A_8 \equiv pq' - p'q - q^2 = 0$$
((I))

,

$$A_5 \equiv cp - bq = 0$$
((II))

.

Lety 0 denote a non-zero solution of a homogeneous differential equation. Then all constant multiples ofy 0 are also solutions of the same equation and form a set {y 0} called a solution ray. A solution ray is uniquely determined by any one of its non-zero members. Different solution rays can have only the zero element in common. Wheny 0 is a non-zero singular solution of equation (ii), all the elements of {y 0} are also singular solutions. In such a case {y 0} is called a singular solution ray.

When the condition (I) is satisfied (and sop≠0) the solution ray of the equationpy′+qy=0 is also a solution ray of equation (ii) and will be called fragmentary. A fragmentary solution ray is singular if and only if the condition (II) is also satisfied. A new indeterminatet (acting effectively as a parameter) will be introduced by algebraic considerations. Equation (ii) then determines a “correlation equation”

$$M(t,y) \equiv (pt^2 - bt + ar)y' + (qt^2 - ct + as)y = 0$$
((iii))

and an “allocation equation”

$$A(t) \equiv (A_5 t^2 + A_5 t + A_1 )t' + (A_8 t^2 + A_6 t^3 + A_4 t^2 + A_2 t + A_0 ) = 0$$
((iv))

. Here, the elementsA i will appear as explicit polynomial expressions in the coefficients of equation (ii). All non-fragmentary solution rays of equation (ii) are then given by the parameterized solution rays {y t 0} of the equationM (t 0,y)=0 corresponding to each solutiont 0 belonging to\(\mathfrak{F}\) of the allocation equation (iv). Only non-zero solutionst 0 of the allocation equation are permitted when the coefficienta is zero.

We shall find that equation (ii) possesses singular solution rays if and only if its corresponding allocation polynomialA(t) is multiplicatively reducible witha≠0. In such a case one of the factors ofA(t) must be an ordinary algebraic polynomial int of the first or second degree. The zeros of this polynomial are precisely those solutions of the allocation equation which give rise to singular parameterized solution rays. There can be no more than two singular solution rays (one of which may be fragmentary).

Equation (i) has been investigated byP. Appell [5] when there exists an element λ for whichQ′(y)+λQ(y) is expressible as the product of two linear homogeneous differential polynomials. In such a case, assuming thatQ(y) is multiplicatively irreducible witha≠0, he has shown that equation (i) possesses two singular solution rays as well as a solution of the formh 2 u 1+hku 2+k 2 u 3 whereh andk are arbitrary constant elements. In our procedure this means that the allocation polynomialA(t) must be multiplicatively reducible. Later we will exhibit a homogeneous quadratic differential equation not satisfying the Appell condition and yet having one singular solution ray (and a multiplicatively reducible allocation polynomial). From a computational standpoint it is much easier to check the factorability of the differential polynomialA(t) than it is to check the Appell condition directly. In fact, even when the procedure ofP. Appell is applicable, it is easier to obtain the solutions by the method presented here.

In the last section we will see examples of homogeneous quadratic differential equations having zero, one, or two singular solution rays. For any integern≧3 there is an equation of type (i) having preciselyn as the maximum number of linearly independent solutions. Also, an equation of type (i) can have infinitely many linearly independent solutions.

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Chalkley, R. On the second order homogeneous quadratic differential equation. Math. Ann. 141, 87–98 (1960). https://doi.org/10.1007/BF01367452

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