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On dual equation in theory of the second order ODE's

The relations between the second order ODE's cubical on the first derivative and their dual equations are discussed

from geometrical point of view was studied by E.Cartan [1].
In fact, according to the expressions on curvature of the space of linear elements (x,y,y') connected with equation (1) where: two types of equations by a natural way are evolved: the first type from the condition a = 0 and second type from the condition b = 0. The first condition a = 0 lead to the the equation in form (1) and the second condition give us the equations (2) where the function g(a, b, b ′ ) satisfies the above p.d.e. (3).
From the elementary point of view the relation between both equations (1) and (2) is a result of the special properties of their General Integral F (x, y, a, b) = 0 which can be considered as the equation of some 3-dim orbifold.

Method of solution and reductions
The equation (2) forming dual pair with some equation (1) can be find from the solutions of the p.d.e. (3).
For solutions of this type of equation we use the method of solution of the p.d.e.'s described first in [3].
To integrate the partial nonlinear first order differential equation can be applied a following approach. We use the following parametric presentation of the functions and variables where variable t is considered as parameter.
Remark that conditions of the type are fulfilled at the such type of presentation.
In result instead of equation (4) one get the relation between the new variables u(x, t, z) and v(x, t, z) and their partial derivatives In some cases the solution of such type of indefinite equation is more simple problem than solution of the equation (4).
The equation (3) has many types of reductions and the simplest of them are For every type of reduction we can write corresponding equation (3) and then integrate it. Remark that the first examples of solutions of equation (3) were obtained in [2][3][4][5][6][7][8].

Proposition 1 Equation (3) can be represent in form
So in standard name of variables we get 3 Two-dimensional short-cut (x, y) equation At the condition h(x, z, y) = 0 we get the equation In particular case f (x, z, y) = f (x, y) the equation takes the form It was integrated in the [4] by the Legendre transformation and another methods in more latest publications.
We consider a new approach to integration of this equation. The change from the equation (10) to the (6) lead to the relation between the functions u(x, t) and v(x, t) and their derivatives 2 The substitutions here of the form give us the linear p.d.e. equation The equation (11) is transformed into the form and can be integrated by the Laplace-method.
In particular case the equation (11) admits the solution with arbitrary parameters Ci and elimination of the parameter t from the relations give us the solution of the equation (10) taking of of the form at the condition C4 = 0 So the equation is dual for the some equation in form (1).
In fact General solution of the equation (12) is defined by the relation and elimination of the variables a, b from here give us the equation in form (1) (13) Remark that the equation (13) is the Rikkati equation with respect of variable z(x) = dy dx .
4 Two-dimensional short-cut (z, y) equation The next example is the equation (9) at the conditions h(x, z, y) = 0 and f (x, z, For a such equation we get the relation A simplest solution of the equation (15) can be find in form where the function A(t) satisfies the linear equation having the solution After inverse transformation in the case C2 = 0 we find corresponding solution of the equation is dual equation for the some equation of the form (1). It can be reduced to the first order ODE In general case C1 = 0, C2 = 0 we also get the first order ODE having singular solution in the form which corresponds the function determined from the equation In more general case the solution of the equation (15) has the form where k is essential parameter. With help of the function ω(z, t) a large class of solutions of the equation (14) radically depending from the choice of parameter k can be produced.

Two-dimensional full (x, y)equation
In the case f (x, z, y) = f (x, y), h(x, z, y) = h(x, y) = 0 from the system (8) we find full (x, y)equation This equation can be transformed into the form ( [5]) with the help of presentation y − Ω(f (x, y), x) = 0.
From the equation (17) we find that the function Ω x (f, x) defined by the relation satisfies the equation with arbitrary coefficients. Af (x, y) 2 + (By 3 + Cy 2 + Ey + F )f (x, y) + Hy 6 + Ky 5 + Ly 4 + My 3 + Ny 2 + P y + Q = 0 and (16) lead to the conditions for determination of the coefficients. As example we find From General Integral we find dual equation Corresponding dual equation looks as The equation has the form On rearrangement we find the relation From here with the help of substitution we get the equation We consider the solutions of the equation (19) in form where k is parameter.
In result we get linear equation for the function A(x, t) It can be transformed into the form with help of the substitutions In result the Laplace-equation has been obtained.
For a given case we have 7 Full (x, y, z)equation The construction of non trivial solutions of a full (x, y, z)-equation may be interested to understanding of the properties of 3-dim orbifolds F (x, y, a, b) = 0 defined by the second order ODE's (1).