Full Text
ON THE NUMBER OF OPERATIONS IN THE INVESTIGATION OF A SINGULAR POINT OF A DIFFERENTIAL EQUATION BY FROMMER’S METHOD
A. F. ANDREEV
Consider the differential equation
\[ \frac{dy}{dx}=\frac{P(x,y)}{Q(x,y)}, \tag{0.1} \]
where \(P\) and \(Q\) are series in integral nonnegative powers of \(x\) and \(y\) with real coefficients, convergent in some neighborhood of the point \(O(0,0)\), \(P(0,0)=Q(0,0)=0\), \(P^2+Q^2>0\) for \(0<x^2+y^2<\rho^2\), \(\rho>0\) a constant.
As is known, the problem of the existence for equation (0.1) of integral curves approaching the point \(O\) with bounded polar angle1 can be investigated by Frommer’s method [1]. In article [2] I undertook a justification of this method.2 In order to formulate the problem of the present note, let us briefly set forth the content of article [2]. It consists in the following.
It is proved that every integral curve of equation (0.1) approaching the point \(O\) with bounded polar angle and not coinciding with any of the coordinate semiaxes, in a sufficiently small neighborhood of the point \(O\), is situated inside one of the coordinate quadrants. The investigation is carried out for integral curves approaching the point \(O\) from the first quadrant—for \(O_1\)-curves (the results are transferred to the remaining quadrants in an obvious way). For an \(O_1\)-curve of equation (0.1), the concepts of the order and the measure of curvature at the point \(O\) are made precise (p. 7). It is proved that every \(O_1\)-curve has a definite order and a definite measure of curvature (Theorems 1 and 5). It is shown (Theorem 2) that the orders of curvature may be \(0\), \(+\infty\), and, besides these, only characteristic and special numbers (p. 10), all of which are positive. Criteria are established for the existence of \(O_1\)-curves with zero and infinite orders of curvature (Theorems 3 and 4). The possible finite orders of curvature are classified into ordinary and special (p. 18). It is shown that a special number is always a special order of curvature, while a characteristic number, as a rule, is an ordinary order, but may also be special (p. 19). It is established that to a special order of curvature there always corresponds a family of \(O_1\)-curves depending on a continuously varying parameter; moreover, to each value of the parameter there corresponds one \(O_1\)-curve, except perhaps for only a finite number of values of this parameter, for each of which the question of the existence and number of \(O_1\)-cur-
curves corresponding to this value requires additional investigation (Theorem 8). Next, the ordinary possible order of curvature \(\nu_0\) is investigated. Criteria are established for the existence of \(O_1\)-curves with order \(\nu_0\) and with zero and infinite measures of curvature (Theorems 6 and 7). It turns out that, besides \(0\) and \(+\infty\), the measures of curvature of \(O_1\)-curves corresponding to the order \(\nu_0\) can only be the nonzero real roots of a certain polynomial (Theorem 9). These possible finite measures of curvature are classified into ordinary and exceptional (pp. 19). It is shown that the question of the existence of \(O_1\)-curves with order \(\nu_0\) and with finite measure of curvature \(\gamma_0\), i.e., \(O_1\)-curves of the form\(^1\)
\[ y=\gamma_0 x^{\nu_0}+o(x^{\nu_0}), \]
is reduced, by means of the substitution
\[ y=(\gamma_0+y_1)x^{\nu_0} \tag{0.2} \]
to the question of the existence of \(O^+\)-curves (\(O\)-curves approaching the point \(O(0,0)\) from the right): a) for the equation
\[ x^l\frac{dy_1}{dx}=R\left(x^{\frac{1}{q}},y_1\right) \tag{0.3} \]
(where \(q\geqslant 1\) is an integer, \(l=\frac{p}{q}\geqslant 1\) is a rational number, \(R(x_1,y_1)\) is a function analytic at the point \((0,0)\) in \(x_1\) and \(y_1\), \(R(0,0)=0\), \(R(0,y_1)\not\equiv 0\)), if \(\gamma_0\) is an ordinary measure; b) for the equation
\[ x^l\frac{dy_1}{dx} = \frac{R\left(x^{\frac{1}{q}},y_1\right)} {S\left(x^{\frac{1}{q}},y_1\right)} \tag{0.4} \]
(where \(q\), \(l\), and \(R(x_1,y_1)\) are the same as in (0.3), and \(S(x_1,y_1)\) is of the same kind as \(R(x_1,y_1)\)), if \(\gamma_0\) is an exceptional measure (pp. 19–20). It is noted (p. 20) that for equation (0.3), in the cases where \(l=1\) or where
\[ R(0,y_1)=a_1y_1+a_2y_1^2+\cdots,\qquad a_1\ne 0, \]
the arrangement of integral curves in a neighborhood of \((0,0)\) is known; and if the substitution (0.2) leads to an equation (0.3) for which neither of these two conditions is satisfied, or to an equation (0.4), then the resulting equation, by means of the substitution \(x=x_1^q\), is brought to the form (0.1), after which the Frommer method can again be applied. In conclusion it is noted that the set of characteristic numbers of equation (0.1) is the same for all coordinate quadrants and, consequently, for the ordinary order of curvature \(\nu_0\), the investigation may be carried out at once at least for the entire right half-plane (for \(O^+\)-curves).
It follows from these results that, for equation (0.1), questions concerning the existence of \(O\)-curves with zero or infinite orders of curvature, as well as with ordinary order and with zero or infinite measures of curvature, can be clarified directly from the form of this equation itself. The question of the existence of \(O\)-curves with any exceptional order of curvature is also, in the main, decided at once, as soon as we have established that this order is exceptional. The question, however, of the existence of \(O\)-curves with ordinary (possible) order and finite measure of curvature may require a transition from equation (0.1)
\(^1\) The symbol \(o(x^{\nu_0})\) denotes an infinitely small quantity of higher order than \(x^{\nu_0}\) (as \(x\to 0\)).
to new equations of the same form and repeated application of the method. In article [2] I did not undertake an investigation of the question of the number of such repetitions (steps) necessary for obtaining an exhaustive answer to the question of interest to us, referring the reader to article [3], in which this question is investigated and an upper estimate is given for the number of the steps mentioned. However, later I discovered that one of the two main theorems of article [3]—Theorem 4, which treats the problem in the general case, is inaccurate (see the example in § 8 of the present note), and that the investigation of the problem of the finiteness of the number of steps in studying the \(TO\)-curves of equation (0.1) by Frommer’s method and of estimating this number thus remains unfinished.
The present note is devoted to the investigation of these questions. In essence it is a continuation of article [2]. The terminology, notation, and results of that article will be used in it. The terminology and some ideas of article [3] are also used. The investigation is carried out for the half-plane \(x>0\). In connection with this, below everywhere the term “\(O\)-curve” is given the following meaning: “an integral curve adjoining the origin from the half-plane \(x>0\) (an \(O^{+}\)-curve).” However, since the final conclusions are formulated in terms of the exponents of the powers of \(x\) and \(y\), they remain valid also for the half-plane \(x<0\).
§ 1. Equation (0.1) can in principle always be written in the form
\[ x^{l}\frac{dy}{dx}=E(x,y)\frac{A_m(x,y)}{B_n(x,y)}, \tag{1.1} \]
where \(l,m\), and \(n\) are integers, \(m\geq 0,\ n\geq 0\); \(E(x,y)\) is a function of its arguments analytic at the point \(x=0,\ y=0\), real for real \(x\) and \(y\), \(E(0,0)\neq 0\) (any function of two variables possessing such properties will be called a function of type \(E\));
\[ A_m(x,y)=y^m+a_1(x)y^{m-1}+a_2(x)y^{m-2}+\cdots+a_m(x), \]
\[ B_n(x,y)=y^n+b_1(x)y^{n-1}+b_2(x)y^{n-2}+\cdots+b_n(x), \]
where \(a_i(x)\) and \(b_j(x)\) are functions of their argument analytic at the point \(x=0\), real for real \(x\), \(a_i(0)=b_j(0)=0\), \(i=1,2,\ldots,m,\ j=1,2,\ldots,n\) (any function of \(x\) and \(y\) having such a structure will be called a special polynomial in \(y\)); \(m\geq 1\), if \(l>1\); \(n\geq 1\), if \(l<-1\); \(mn\geq 1\), if \(l=0\). In what follows we shall take into account only those of the functions \(a_i(x)\), \(b_j(x)\) which are not identically zero. These functions can be represented in the form
\[ a_i(x)=a_{ik_i}x^{k_i}+o(x^{k_i}),\quad k_i\geq 1,\quad a_{ik_i}\neq 0,\quad i\overset{0}{=}1,2,\ldots,m, \]
\[ b_j(x)=b_{jl_j}x^{l_j}+o(x^{l_j}),\quad l_j\geq 1,\quad b_{jl_j}\neq 0,\quad j\overset{0}{=}1,2,\ldots,n, \]
where the symbol \(i\overset{0}{=}1,2,\ldots,m\) means that the index \(i\) assumes all integer values from \(1\) to \(m\), excluding each value \(i=i_0\) such that \(a_{i_0}(x)\equiv 0\) (the symbol \(j\overset{0}{=}1,2,\ldots,n\) is defined analogously).
Definition. Let \(q\geq 1\) be an integer. If the functions \(a_i(x)\), \(i=1,2,\ldots,m\), and \(b_j(x)\), \(j=1,2,\ldots,n\), are expanded in series in integral
positive powers of the quantity \(x^{q!}\), the function \(E(x,y)\) can be expanded in a series in integral nonnegative powers of \(x^{q!}\) and \(y\), and the number \(l \equiv 1(\operatorname{mod}(q!))\), then we shall say that equation (1.1) has the (!)-property (factorial-property) with respect to the number \(q\).
It is obvious that equation (1.1) always has the (!)-property with respect to the number \(q=1\).
Lemma 1. If in equation (1.1) one makes the substitution \(x=x_1^{q!}\), where \(q>1\) is an integer, then one obtains an equation of the same form, possessing the (!)-property with respect to the number \(q\).
The validity of this assertion is obvious.
Lemma 2. If 1) equation (1.1) has the (!)-property with respect to the number \(q\), 2) \(p\) is an integer, \(1 \le p \le q\), 3) \(\nu \ge 1\) is an integer, \(\nu \equiv 0(\operatorname{mod}(p!))\), 4) \(\gamma\) is a real number, then the substitution
\[ y=(\gamma+y_1)x^\nu \tag{1.2} \]
transforms equation (1.1) into an equation of the same form, possessing the (!)-property with respect to the number \(p\).
Proof. Substituting (1.2) into (1.1), we obtain
\[ x^{l+\nu}\frac{dy_1}{dx} = [E(x,(\gamma+y_1)x^\nu)A_m(x,(\gamma+y_1)x^\nu)- \]
\[ -\nu(\gamma+y_1)x^{l+\nu-1}B_n(x,(\gamma+y_1)x^\nu)] [B_n(x,(\gamma+y_1)x^\nu)]^{-1}. \tag{1.3} \]
Here the numerator and denominator of the right-hand side are series in integral nonnegative powers of \(x^{p!}\) and \(y_1\), convergent for small \(x\) and \(y_1\), and \(\nu+l \equiv 1(\operatorname{mod}(p!))\). Consequently, after reducing this equation to the form (1.1), it will have the (!)-property with respect to \(p\). The lemma is proved.
Lemma 3. Let \(C_n(x,y)\) be a special polynomial in \(y\) of degree \(n \ge 1\), whose coefficients \(c_i(x)\), \(i=1,2,\ldots,n\), can be expanded in real series in integral positive powers of the quantity \(x^{n!}\), convergent for sufficiently small \(|x|\). Then
\[ C_n(x,y)\equiv \prod_{i=1}^{n}(y-\chi_i(x)), \]
where \(\chi_i(x)\), \(i=1,2,\ldots,n\), are series in integral positive powers of \(x\) (generally speaking, with complex coefficients), convergent for sufficiently small \(|x|\).
Proof. By assumption
\[ C_n(x,y)\equiv D_n(x^{n!},y), \]
where \(D_n(x_1,y)\) is a special polynomial in \(y\). But, as is known from the theory of algebraic functions (see, for example, [4], p. 670),
\[ D_n(x_1,y)\equiv \prod_{i=1}^{n}\left(y-\eta_i\left(x_1^{\frac{1}{p_i}}\right)\right), \tag{1.4} \]
where \(p_i\) are integers, \(1 \le p_i \le n\), and \(\eta_i(t)\) are series in integral positive powers of \(t\) (generally speaking, with complex coefficients), converging in some neighborhood of the point \(t=0\) \((i=1,2,\ldots,n)\). Putting \(x_1=x^{n!}\) in (1.4), we obtain the assertion of the lemma.
Let \(R(x)\) be the resultant of the polynomials \(A_m(x,y)\) and \(B_n(x,y)\). As is known ([5], pp. 233–235),
\[ R(x)= \left| \begin{array}{cccccc} \multicolumn{6}{c}{\left.\vphantom{\begin{array}{c}1\\1\\ \cdots\\1\end{array}}\right\} n\ \text{rows}}\\[-1.5ex] 1 & a_1(x) & \cdots & a_{m-1}(x) & a_m(x) \\ & 1 & a_1(x) & \cdots & a_{m-1}(x) & a_m(x) \\ & & \cdots & \cdots & \cdots & \cdots \\ & & 1 & a_1(x) & \cdots & a_{m-1}(x) & a_m(x) \\ \multicolumn{6}{c}{\left.\vphantom{\begin{array}{c}1\\1\\ \cdots\\1\end{array}}\right\} m\ \text{rows}}\\[-1.5ex] 1 & b_1(x) & \cdots & b_{n-1}(x) & b_n(x) \\ & 1 & b_1(x) & \cdots & b_{n-1}(x) & b_n(x) \\ & & \cdots & \cdots & \cdots & \cdots \\ & & 1 & b_1(x) & \cdots & b_{n-1}(x) & b_n(x) \end{array} \right| \tag{1.5} \]
(in the vacant places there are zeros), and formula (1.5) may be regarded as valid also in the case when one of the numbers \(m, n\) is equal to zero (under our assumptions concerning equation (1.1), both of them cannot be zero). It is obvious that \(R(x)\) is an analytic function of \(x\) at the point \(x=0\), and \(R(x)\not\equiv 0\). Extracting the principal term, it may be written in the form
\[ R(x)\equiv ax^r+o(x^r),\quad r\geq 0,\quad a\neq 0\text{ is a constant}. \tag{1.6} \]
If one of the numbers \(m,n\) is equal to zero, then, as follows from (1.5), \(R(x)\equiv 1\), and consequently \(r=0\). If \(m\geq 1\) and \(n\geq 1\), then
\[ R(x)\equiv \prod_{i,j=1}^{m,n}\bigl(\varphi_i(x)-\psi_j(x)\bigr), \]
where \(\varphi_i(x)\) and \(\psi_j(x)\) are the roots of the polynomials \(A_m(x,y)\) and \(B_n(x,y)\), respectively. Since
\[ \varphi_i(0)=\psi_j(0)=0,\quad i=1,2,\ldots,m,\ j=1,2,\ldots,n, \]
in this case \(r\geq 1\). By assumption,
\[ \varphi_i(x)\not\equiv \psi_j(x),\quad i=1,2,\ldots,m,\ j=1,2,\ldots,n. \]
Therefore, for arbitrary \(i\) and \(j\), one may put
\[ \varphi_i(x)-\psi_j(x)=a_{ij}x^{l_{ij}}+o\bigl(x^{l_{ij}}\bigr), \tag{1.7} \]
where each number \(a_{ij}\neq 0\), and each number \(l_{ij}>0\).
Lemma 4. If in equation (1.1) \(mn\geq 1\), and if it possesses property (!) with respect to the number \(K\),
\[ K=\max\{m,n\}, \tag{1.8} \]
then the exponents \(r\) and \(l_{ij}\), determined by formulas (1.6) and (1.7), satisfy the inequalities
\[ r\geq mn,\quad 1\leq l_{ij}\leq r-mn+1,\quad i=1,2,\ldots,m,\ j=1,2,\ldots,n. \]
Proof. Under the assumptions of Lemma 4, the polynomials \(A_m(x,y)\) and \(B_n(x,y)\) satisfy the conditions of Lemma 3, and therefore \(\varphi_i(x)\) and \(\psi_j(x)\) are series in integral positive powers of \(x\). Hence the assertion of the present lemma follows.
Lemma 5. If equation (1.1) possesses property (!) with respect to the number \(K\), then equation (1.1)
1) in the case \(n=0\) is representable in the form
\[ x^l \frac{dy}{dx}=E(x,y)\prod_{i=1}^{m}\bigl(y-\varphi_i(x)\bigr),\quad l\geqslant 1; \tag{1.9} \]
2) in the case \(m=0\) it can be represented in the form
\[ x^l \frac{dy}{dx}=\frac{E(x,y)}{\displaystyle\prod_{j=1}^{n}\bigl(y-\psi_j(x)\bigr)},\quad l\leqslant -1; \tag{1.10} \]
3) in the case \(mn\geqslant 1\) it can be represented in the form
\[ x^l \frac{dy}{dx} = E(x,y) \frac{\displaystyle\prod_{i=1}^{m}\bigl(y-\varphi_i(x)\bigr)} {\displaystyle\prod_{j=1}^{n}\bigl(y-\psi_j(x)\bigr)}, \tag{1.11} \]
where \(\varphi_i(x)\) and \(\psi_j(x)\) are series in integral positive powers of \(x\), generally speaking, with complex coefficients.
This assertion is also a consequence of Lemma 3.
§ 2. Consider the characteristic polygon of equation (1.1) ([2], pp. 8–10). It may be constructed in the following way. In the auxiliary \(\nu\mu\)-plane we construct the straight lines
\[ \mu=(m-i)\nu+k_i+1,\quad i\overset{0}{=}0,1,2,\ldots,m, \tag{2.1} \]
\[ \mu=(n-j+1)\nu+l_j+l,\quad j\overset{0}{=}0,1,2,\ldots,n \tag{2.2} \]
(\(k_0=l_0=0\)). The totality of these straight lines will be called the Frommer diagram for equation (1.1). In the half-plane \(\nu>0\), from the segments of these straight lines we construct a polygonal line so that below this polygonal line (i.e. for smaller \(\mu\)) there are no points of the straight lines (2.1), (2.2). This will be the characteristic polygonal line of equation (1.1). We shall agree to call any side of the characteristic polygonal line lying on one of the straight lines (2.1) a side of the first class, and any side lying on one of the straight lines (2.2) a side of the second class. In this case any side of a given class may turn out to be double (i.e. belonging also to the other class), unless it is specifically stipulated that it is simple. From the exponents \(l,m,n\) of equation (1.1) we define the number \(L\) by the following formula:
\[ L= \begin{cases} m, & \text{if } l\geqslant 1,\\ n+1, & \text{if } l\leqslant 0. \end{cases} \tag{2.3} \]
Lemma 6. The angular coefficients of the sides of the characteristic polygonal line do not exceed \(L\). The number of sides does not exceed \(L+1\).
Proof. Let \(l\geqslant 1\). Then the angular coefficients of the straight lines (2.1) do not exceed \(L\), while for the straight lines (2.2) \(l_j+l\geqslant l\geqslant 1\), \(j\overset{0}{=}0,1,\ldots,n\), and therefore those of them whose angular coefficients \(n-j+1>L=m\) are situated, for \(\nu>0\), above the straight line \(\mu=m\nu+1\) of class (2.1) and, consequently, do not participate in the formation of the characteristic polygonal line. Let \(l\leqslant 0\). Then the angular coefficients of the stra-
of the lines (2.2) do not exceed \(L\), while for the lines (2.1) \(k_i+1\geqslant 1>l,\ i=0, 1, 2,\ldots,m\), and therefore those of them for which \(m-i\geqslant L=n+1\) are, for \(v>0\), situated above the line \(\mu=(n+1)v+l\) of class (2.2) and do not participate in the formation of the characteristic polygonal line. The first assertion of the lemma is proved. The second follows from the first, if one takes into account that the angular coefficients of the links of the characteristic polygonal line are nonnegative integers (and moreover distinct).
Corollary. Equation (1.1) has no more than \(L\) characteristic numbers.
This assertion is obvious, since the characteristic numbers are the abscissas of the vertices of the characteristic polygonal line.
For a characteristic number \(v_0\) of equation (1.1) only the following three possibilities can occur:
1) \(v_0\) is the abscissa of the point of intersection of two links of the first class of the characteristic polygonal line; in this case
\[ v_0=\frac{k_{i_2}-k_{i_1}}{i_2-i_1},\quad 1\leqslant i_2-i_1\leqslant L,\quad k_{i_2}>k_{i_1}; \tag{2.4} \]
2) \(v_0\) is the abscissa of the point of intersection of two links of the second class of the characteristic polygonal line; in this case
\[ v_0=\frac{l_{j_2}-l_{j_1}}{j_2-j_1},\quad 1\leqslant j_2-j_1\leqslant L,\quad l_{j_2}>l_{j_1}; \tag{2.5} \]
3) \(v_0\) is the abscissa of the point of intersection of links of the first and second classes of the characteristic polygonal line; in this case
\[ v_0=\frac{|\,l_{j_0}-k_{i_0}+l-1\,|}{|(m-i_0)-(n-j_0+1)|}, \tag{2.6} \]
\[ 1\leqslant |(m-i_0)-(n-j_0+1)|\leqslant L,\quad |\,l_{j_0}-k_{i_0}+l-1\,|\geqslant 1. \]
From formulas (2.4), (2.5), (2.6) it is seen that any characteristic number \(v_0\) of equation (1.1) can be represented in the form \(v_0=\dfrac{p}{q}\), where \(p\) and \(q\) are positive integers, and moreover \(1\leqslant q\leqslant L\).
Let \(v_0=\dfrac{p}{q}\) be a characteristic number of equation (1.1), which is an ordinary (possible) order of curvature of the \(O\)-curves of equation (1.1). Then the substitution \(y=ux^{v_0}\) brings equation (1.1) to the form ([2], p. 15)
\[ x^{l_1}\frac{du}{dx}= \frac{P_1(u)+x^{\lambda_1}R_1(x,u)} {P_2(u)+x^{\lambda_2}R_2(x,u)}, \tag{2.7} \]
where \(l_1,\lambda_1\), and \(\lambda_2\) are positive rational numbers with common denominator \(q\), \(l_1\geqslant 1\), \(P_1(u)\) and \(P_2(u)\) are polynomials in \(u\) with constant coefficients, \(P_1(u)\ne0\), \(P_2(u)\ne0\), and \(R_1(x,u)\) and \(R_2(x,u)\) are series in nonnegative integral powers of \(x^{1/q}\) and \(u\), convergent in the region
\[ 0\leqslant x<\alpha<1,\quad |u|<\frac{\beta}{x^\gamma}, \]
\(\alpha\) and \(\beta\) being positive constants.
Let us examine the structure of the polynomials \(P_1(u)\) and \(P_2(u)\). The exponents of \(u\) in the terms of the polynomial \(P_1(u)\) are the angular coefficients
of the lines (2.1), (2.2) passing through the vertex of the characteristic polygonal line with the abscissa under consideration \(\nu_0\). Therefore, writing \(P_1(u)\) in the form
\[ P_1(u)=u^\rho(p_0u^s+p_1u^{s-1}+\ldots+p_s),\quad p_0p_s\ne 0,\quad \rho\geq 0 \tag{2.8} \]
and taking into account that, for this polynomial, the terms corresponding to the double lines of Frommer’s diagram may vanish, we shall have \(0\leq s\leq L\).
The exponents of \(u\) in the terms of the polynomial \(P_2(u)\) are the angular coefficients of the lines (2.2). Therefore, putting
\[ P_2(u)=u^\tau(q_0u^\sigma+q_1u^{\sigma-1}+\ldots+q_\sigma),\quad q_0q_\sigma\ne 0,\quad \tau\geq 0, \]
we shall have \(0\leq \sigma\leq n\).
The finite (distinct from \(0\) and \(\infty\)) measures of curvature of the \(O\)-curves of equation (1.1) corresponding to the ordinary order of curvature \(\nu_0\) can only be the nonzero real roots of the polynomial \(P_1(u)\).
Suppose in (2.8) \(s\geq 1\), and let \(\gamma_0\ne 0\) be a root of the polynomial \(P_1(u)\) of multiplicity \(m_1\), \(1\leq m_1\leq s\). Let \(n_1\) be the multiplicity of \(\gamma_0\) as a root of the polynomial \(P_2(u)\), \(n_1\geq 0\). Then the substitution \(u=\gamma_0+y_1\) transforms equation (2.7) into the equation
\[ x^{l_1}\frac{dy_1}{dx} = E_1\left(x^{\frac{1}{q}},\,y_1\right) \frac{ A^{(1)}_{m_1}\left(x^{\frac{1}{q}},\,y_1\right) }{ B^{(1)}_{n_1}\left(x^{\frac{1}{q}},\,y_1\right) }, \tag{2.9} \]
where \(l_1\) is the same as in equation (2.7), \(E_1(x_1,y_1)\) is a function of type \(E\), and \(A^{(1)}_{m_1}(x_1,y_1)\) and \(B^{(1)}_{n_1}(x_1,y_1)\) are special polynomials in \(y_1\).
Following [3], we shall call equation (2.9), obtained as a result of the substitution
\[ y=(\gamma_0+y_1)x^{\nu_0}, \tag{2.10} \]
from equation (1.1), a derived equation of equation (1.1). Each pair \(\nu,\gamma\)—where \(\nu\) is an ordinary (possible) order of curvature of the \(O\)-curves of equation (1.1), and \(\gamma\ne 0\) is the corresponding (possible) measure of curvature—generates its own substitution (2.10) and, consequently, its own derived equation (2.9). All these equations form the set of derived equations of the first series (from equation (1.1)).
Lemma 7. Equation (1.1) has \(\omega_1\), \(0\leq \omega_1\leq L\), derived equations of the first series (these equations have the form (2.9)). Moreover, if the numbers \(m_1^{(\alpha)}\), \(\alpha=1,2,\ldots,\omega_1\), play in these equations the role of the exponent \(m_1\), then
\[ \sum_{\alpha=1}^{\omega_1} m_1^{(\alpha)} \leq L. \]
Proof. If equation (1.1) has no characteristic numbers, then it has no derived equations either. In this case the assertion of the lemma is true. Suppose equation (1.1) has characteristic numbers. According to the corollary to Lemma 6, their number does not exceed \(L\). Let \(p\) of them, \(1\leq p\leq L\), be ordinary (possible) orders of curvature of the \(O\)-curves of equation (1.1). Let these be the numbers \(\nu^{(1)},\nu^{(2)},\ldots,\nu^{(p)}\) \((\nu^{(1)}<\nu^{(2)}<\ldots<\nu^{(p)}\), if \(p>1)\), and let the polynomials \(P_1^{(1)}(u), P_1^{(2)}(u),\ldots,P_1^{(p)}(u)\) play the role of the polynomial \(P_1(u)\) in the equations corresponding to these numbers.
... of the form (2.7). If none of these polynomials has nonzero real roots, then equation (1.1) has no derivative equations and the assertion of the lemma is true. Suppose that at least one of the polynomials \(P_1^{(i)}\) has nonzero real roots. Let \(\gamma_j^{(i)}\), \(j = 1, 2, \ldots, \omega^{(i)}\), be the distinct nonzero real roots of the polynomial \(P_1^{(i)}(u)\), \(i = 1, 2, \ldots, p\) (for those values of \(i\) for which the set \(\{\gamma_j^{(i)}\}\) is empty, we shall put \(\omega^{(i)} = 0\)). The totality of derivative equations of the first series for (1.1) is in one-to-one correspondence with the set of ordered pairs of numbers \(\nu^{(i)}, \gamma_j^{(i)}\), \(i = 1, 2, \ldots, p\), \(j = 1, 2, \ldots, \omega^{(i)}\). Therefore, if \(\omega_1\) is the number of such equations, then in the case under consideration
\[ \omega_1 = \sum_{i=1}^{p} \omega^{(i)} . \]
But the structure of the polynomials \(P_1^{(i)}(u)\) is such that the exponent of the lowest degree of \(P_1^{(i)}(u)\) is not less than the exponent of the highest degree of \(P_1^{(i+1)}(u)\), \(i = 1, 2, \ldots, p - 1\), and the degree of \(P_1^{(1)}(u)\) (according to Lemma 6) does not exceed \(L\). Thus, representing these polynomials in the form (2.8), we obtain
\[ P_1^{(i)}(u) = u^{\rho_i}\left(p_0^{(i)}u^{s_i} + p_1^{(i)}u^{s_i-1} + \ldots + p_{s_i}^{(i)}\right), \]
\[ i = 1, 2, \ldots, p, \qquad s_1 + s_2 + \ldots + s_p \leq L. \]
Consequently, \(\omega^{(i)} \leq s_i\), \(i = 1, 2, \ldots, p\), and therefore the number of derivative equations of the first series for (1.1) is
\[ \omega_1 = \sum_{i=1}^{p} \omega^{(i)} \leq \sum_{i=1}^{p} s_i \leq L. \]
Further, the numbers \(m_1^{(\alpha)}\) in these equations are the multiplicities of the roots \(\gamma_j^{(i)}\) of the polynomials \(P_i^{(i)}(u)\). Therefore
\[ \sum_{\alpha=1}^{\omega_1} m_1^{(\alpha)} \leq \sum_{i=1}^{p} s_i \leq L. \]
The lemma is proved.
Remark. Assertions equivalent to those of Lemmas 3, 6, and 7 are contained in paper [3]. The first of them is accompanied by a very complicated proof. The other two appear under the heading “easy to prove.”
§ 3. Definition. If equation (1.1) possesses the \((!)\)-property with respect to the number \(L\) defined by formula (2.3), then we shall say that it possesses the \((!)\)-property with respect to the exponents \(l\), \(m\), and \(n\) (or, simply, that it possesses the \((!)\)-property).
Theorem 1. If equation (1.1) possesses the \((!)\)-property, then all its characteristic numbers are integers.
Proof. Let \(\gamma_0\) be an arbitrary characteristic number of equation (1.1). Then it is represented by one of the fractions (2.5), (2.6), (2.7). But in each of these fractions the numerator is divisible by \(L!\) (by assumption), while the denominator does not exceed \(L\). The theorem is proved.
Corollary. If equation (1.1) possesses the \((!)\)-property, then any of its derivative equations of the first series has the form
\[ x^{l_1}\frac{dy_1}{dx} = E_1(x, y_1) \frac{A_{m_1}^{(1)}(x, y_1)}{B_{n_1}^{(1)}(x, y_1)} \tag{3.1} \]
(where \(l_1 \geqslant 1\) is an integer, \(E_1(x,y_1)\) is a function of type \(E\), and \(A_{n_1}^{(1)}(x,y_1)\) and \(B_{n_1}^{(1)}(x,y_1)\) are special polynomials in \(y_1\)), i.e. it is an equation of the form (1.1).
Indeed, if equation (1.1) has the (!)-property, then, as follows from Theorem 1, in any derived equation of the 1st series (2.9) \(q=1\), and, consequently, this equation has the form (3.1).
Everything said in §§ 1, 2 concerning equation (1.1) is applicable to equation (3.1). In particular, generally speaking, it has derived equations. We shall call these equations (following [3]) derived equations of the 2nd series from equation (1.1). If they too admit derived equations, then we shall call the latter derived equations of the 3rd series from equation (1.1), and so on. We shall agree to call the transition from the \((i-1)\)-st series of derived equations (from (1.1)) to the \(i\)-th series of such equations an operation, \(i=1,2,3,\ldots\) (the zeroth series of derived equations from (1.1) consists of the single equation (1.1)). It is evident that the implementation of each operation reduces to carrying out a series of substitutions of the form
\[ y_{i-1}=\left(\gamma_{i-1}^{(j)}+y_i\right)x^{\nu_{i-1}^{(j)}}, \]
where \(i\) is the number of the operation, and \(j\) is the number of the substitution in the given operation.
An important feature of derived equations is their ability to inherit the (!)-property. The proof of this assertion is based on the following lemma.
Lemma 8. If equation (1.1) has the (!)-property, and equation (3.1) is derived from (1.1), obtained from (1.1) by means of the substitution (2.10), then
\[ \nu_0 \equiv 0 \pmod{(m_1!)}. \]
Proof. The number \(m_1\) is the multiplicity of the root \(\gamma_0\) of the polynomial \(P_1(u)\). Therefore, \(1 \leqslant m_1 \leqslant s \leqslant L\) (see (2.8)).
1) \(m_1=s\). Then
\[ P_1(u)=p_0u^\rho(u-\gamma)^s \]
contains all consecutive powers of \(u\) from \(u^\rho\) to \(u^{\rho+s}\) (i.e. at least two consecutive powers of \(u\): \(u^\rho\) and \(u^{\rho+1}\)). Consequently, \(\nu_0\) is the abscissa of a vertex of the characteristic polygon through which pass at least two of the lines (2.1), (2.2), whose slopes differ by exactly one. But then \(\nu_0\) is representable in the form (2.5), (2.6), or (2.7) with denominator 1 and therefore \(\nu_0\) is divisible by \(L!\), where \(L\) is defined by formula (2.3). But \(m_1 \leqslant L\). Therefore all the more \(\nu_0\) is divisible by \(m_1!\).
2) \(1 \leqslant m_1 < s\). Among the lines of the Frommer diagram passing through the vertex of the characteristic polygon with the abscissa under consideration \(\nu_0\), the largest and the smallest slopes have lines carrying sides of the characteristic polygon. These may be
a) two lines of class (2.1): with \(i=i_1\) and \(i=i_2\), \(i_1<i_2\);
b) two lines of class (2.2): with \(j=j_1\) and \(j=j_2\), \(j_1<j_2\);
c) a line of class (2.1) with \(i=i_0\) and a line of class (2.2) with \(j=j_0\), \(m=-i_0+n-j_0+1\) (where in any of these cases one or both of the mentioned lines may be duplicated by lines of the other class).
In case a), on the one hand, \(s \leqslant i_2-i_1\) and, consequently, \(m_1 \leqslant i_2-i_1-1\), while, on the other hand, \(\nu_0\) is representable by formula (2.5) and therefore is divisible by \((i_2-i_1-1)!\). Thus, in this case
\[ \nu_0 \equiv 0 \pmod{(m_1!)}. \]
In cases b) and c) we arrive at the same result by considering instead ofאַס
the differences \(i_2-i_1\) of the quantities \(j_2-j_1\) and \(|(n-j_0+1)-(m-i_0)|\), respectively. The lemma is proved.
Theorem 2. If equation (1.1) has the (!)-property (with respect to the exponents \(l, m\), and \(n\)), then any derivative equation (3.1) of it also has the (!)-property (with respect to the exponents \(l_1, m_1, n_1\)).
Proof. In equation (3.1), \(l_1 \ge 1\). Therefore for it the (!)-property (with respect to \(l_1, m_1\), and \(n_1\)) is the (!)-property with respect to the number \(L_1=m_1\le L\). But equation (3.1) is obtained from (1.1) by a substitution of the form (2.10), and equation (1.1) has the (!)-property with respect to \(L\) (by hypothesis), while \(v_0\equiv 0 \pmod{(L_1!)}\) (by Lemma 8). Thus all the conditions of Lemma 2 are fulfilled, and the assertion of the theorem follows.
Corollary 1. If equation (1.1) has the (!)-property, and equation (3.1) is a derivative of (1.1) of the first series, then the characteristic numbers of equation (3.1) are integers.
Corollary 2. If equation (1.1) has the (!)-property and admits derivative equations of series \(k\ge 1\), then any one of them has the form
\[ x^{l_k}\frac{dy_k}{dx} = E_k(x,y_k) - \frac{A_{m_k}^{(k)}(x,y_k)}{B_{n_k}^{(k)}(x,y_k)} \tag{3.2} \]
(where \(l_k\ge 1\) is an integer, \(E_k(x,y_k)\) is a function of type \(E\), \(A_{m_k}^{(k)}(x,y_k)\), \(B_{n_k}^{(k)}(x,y_k)\) are special polynomials in \(y_k\)), i.e. it is an equation of the form (1.1). Moreover, equation (3.2) also has the (!)-property (with respect to the exponents \(l_k, m_k, n_k\)) and, consequently, its characteristic numbers are integers.
These assertions follow from Theorems 1 and 2.
Lemma 9. Equation (3.2) has no more than \(m_k\) derivative equations of the first series. Moreover, the sum of the exponents \(m_{k+1}^{(\alpha)}\) (playing in these equations the role of the exponent \(m_{k+1}\)) does not exceed \(m_k\).
This assertion follows immediately from Lemma 7, if one takes into account that for equation (3.2) the role of the number \(L\) is played by the number \(L_k=m_k\).
Theorem 3. If equation (1.1) has the (!)-property and \(\omega_k\) is the number of derivative (from (1.1)) equations of the given series \(k\) (these equations have the form (3.2)), then \(0\le \omega_k\le L\). Moreover, if \(m_k^{(\alpha)}\), \(\alpha=1,2,\ldots,\omega_k\), play in these equations the role of the exponent \(m_k\), then
\[ \sum_{\alpha=1}^{\omega_k} m_k^{(\alpha)} \le L. \]
This assertion is easily proved by complete induction: the (!)-property ensures the invariance of the structure of derivative equations, Lemma 7 ensures the base of the induction, and Lemma 9 ensures the possibility of the inductive transition from \(k_0\) to \(k_0+1\).
Definition. If equation (1.1) has the (!)-property with respect to the number \(M\),
\[ M=\max\{K,L\}= \begin{cases} \max\{m,n\}, & \text{if } l\ge 1,\\ \max\{m,n+1\}, & \text{if } l\le 0, \end{cases} \tag{3.3} \]
then we shall say that it has the full (!)-property with respect to the exponents \(l, m\), and \(n\) (or simply that it has the full (!)-property).
Obviously, if equation (1.1) has the full (!)-property, then it 1) has the (!)-property with respect to \(K\), 2) has the (!)-prop-
with respect to \(L\), i.e., the \((!)\)-property (with respect to the indices \(l, m, n\)).
Theorem 4. If equation (1.1) has the full \((!)\)-property (with respect to the indices \(l, m\), and \(n\)), then any derived equation of any series \(k\) also has the full \((!)\)-property (with respect to the corresponding indices \(l_k, m_k\), and \(n_k\)).
This assertion is a generalization, to the case of the full \((!)\)-property, of the assertion obtained by combining Theorem 2 with its second corollary. Its proof may be given according to the same scheme. Only the derivation of the analogue of Lemma 8,
\[ \left(v_0 \equiv 0 \pmod{(M_1!)}\right), \quad M_1=\max\{m_1,n_1\}, \]
has to be divided into four cases: 1) \(m_1=s\), 2) \(n_1=\sigma \geqslant 1\), 3) \(m_1<s,\ 1\leq n_1<\sigma\), 4) \(m_1<s,\ n_1=0\).
§ 4. Suppose that in equation (1.1) \(n=0\). Then it has the form
\[ x^l \frac{dy}{dx}=E(x,y)A_m(x,y), \quad l\geqslant 1,\quad m\geqslant 1. \tag{4.1} \]
Equation (4.1) has been studied in [6, 3, 7]. Nevertheless (for the sake of completeness) I shall carry out its investigation here (from my own point of view).
We shall call equation (4.1) a generalized Briot and Bouquet equation [6]. We agree to distinguish two cases: 1) \(m=1\) (in this case, following [6], we shall call equation (4.1) a simple Briot and Bouquet equation), 2) \(m\geqslant 2\).
In the first case the right-hand side of (4.1) contains the linear term \(E(0,0)y\), and therefore the behavior of the integral curves of this equation in a neighborhood of the point \(O(0,0)\) is clarified elementarily (for example, by the method of constructing Frommer normal domains [8], pp. 102–122, [9]). If \(E(0,0)>0\), an infinite set of \(O\)-curves adjoins the point \(O(0,0)\) from the right; if \(E(0,0)<0\), there is only one \(O\)-curve. The orders and measures of curvature of these curves at the point \(O\) are easily determined (see, for example, [10]). We shall assume that the investigation of the \(O\)-curves of a simple Briot and Bouquet equation does not require operations, i.e., does not require passing to derived equations. Suppose now that in equation (4.1) \(m\geqslant 2\). Suppose, moreover, that it has the \((!)\)-property (which, for (4.1), is the \((!)\)-property with respect to the number \(m\)).
Only the following three possibilities can occur:
1) Equation (4.1) has no ordinary (possible) order of curvature \(v\) to which a finite measure of curvature \(\gamma\) would correspond.
2) Equation (4.1) has at least one such order of curvature \(v\), and any finite measure of curvature \(\gamma\) corresponding to this order is a simple root of the corresponding polynomial \(P_1(u)\).
3) Equation (4.1) has at least one ordinary (possible) order of curvature \(v\), to which there corresponds at least one finite measure of curvature \(\gamma\) that is a multiple root of the corresponding polynomial \(P_1(u)\).
In the first case, the problem of the existence for equation (4.1) of \(O\)-curves with all possible orders of curvature is solved (as follows from the results of article [2]) directly from the form of equation (4.1). Equation (4.1) has no derived equations (in the sense of the definition given in § 2).
In the second case, for equation (4.1) the same \(O\)-curves as in the first case are possible, and, in addition, \(O\)-curves of the form
\[ y=\gamma x^\nu+o(x^\nu),\quad \nu>0\ \text{and}\ \gamma\ne0\ \text{constants}, \tag{4.2} \]
and the question of whether equation (4.1) has \(O\)-curves of the form (4.2), corresponding to fixed \(\nu\) and \(\gamma\), is reduced, by means of the substitution
\[ y=(\gamma+y_1)x^\nu \tag{4.3} \]
to the question of the existence of \(O\)-curves for a derived equation of (4.1) of the form (3.1), in which \(n_1=0,\ m_1=1\). This latter equation will therefore be a simple Briot–Bouquet equation.
In the third case, equation (4.1) may have the same \(O\)-curves as in the 2nd case, and, in addition, \(O\)-curves of the form (4.2), the question of whose existence (for fixed \(\nu\) and \(\gamma\)) is reduced by means of the substitution (4.3) to the question of the existence of \(O\)-curves for a derived equation of (4.1) of the form (3.1), in which \(n_1=0,\ m_1\ge2\). This latter equation will thus have the form of the original equation (and moreover, by Theorem 2, will possess the (!)-property). For it the same three possibilities may arise.
We shall show that this process cannot continue indefinitely, namely, we shall show that for the equations of series \(k\ge1\) derived from (4.1), for sufficiently large \(k\), the third possibility is not realized.
Theorem 5. If the generalized Briot–Bouquet equation (4.1) has exponent \(m\ge2\), possesses the (!)-property, and admits derived equations up to the \(n\)-th series inclusive,
\[ n=\frac{l+1}{2}, \tag{4.4} \]
then any of its derived equations of the \(n\)-th series will be a simple Briot–Bouquet equation.
Proof. By assumption, equation (4.1) has derived equations up to series \(n\ge1\) inclusive. Therefore it admits an ordinary (possible) order of curvature \(\nu_0\), to which there corresponds a finite measure of curvature \(\gamma_0\). Transforming equation (4.1) by means of the substitution \(y=(\gamma_0+y_1)x^{\nu_0}\), we obtain the equation
\[ x^{l+\nu_0}\frac{dy_1}{dx} = E\bigl(x,(\gamma_0+y_1)x^{\nu_0}\bigr) A_m\bigl(x,(\gamma_0+y_1)x^{\nu_0}\bigr) -\nu_0(\gamma_0+y_1)x^{l+\nu_0-1}, \tag{4.5} \]
which (as is not hard to see) can be represented in the form
\[ x^{l_1}\frac{dy_1}{dx}=E_1(x,y_1)A_{m_1}^{(1)}(x,y_1), \tag{4.6} \]
where \(l_1\ge1,\ m_1\ge1,\ E_1(x,y_1)\) is a function of type \(E\), and \(A_{m_1}^{(1)}(x,y_1)\) is a special polynomial in \(y_1\). Equation (4.6) is a derived equation of the 1st series from (4.1). By Theorem 2 it possesses the (!)-property, and, by Lemma 8, the number \(m_1!\) is a divisor of the number \(\nu_0\).
Only the following two possibilities can arise:
1) In equation (4.6), \(m_1=1\). In this case it is a simple Briot–Bouquet equation. It is easy to show that if it has derived equations, then all of them will likewise be simple Briot–Bouquet equations.
2) In equation (4.6), \(m_1\ge2\), i.e., it has the same form as the original equation. We shall show that in this case \(l_1\le l-2\). To this end consider the intermediate equation between (4.1) and (4.6), of the form (2.7),
obtained from (4.1) as a result of the substitution \(y=ux^{\nu_0}\). From the condition \(m_1>2\) it follows that the polynomial \(P_1(u)\) in this equation consists of no fewer than three terms. This means that through the vertex of the characteristic polygonal line of equation (4.1) with the given abscissa \(\nu_0\) there pass no fewer than three lines (2.1), (2.2) with different angular coefficients. Consequently, the angular coefficient of at least one of these lines is not less than 2. Therefore, if \(h\) is the order of smallness with respect to \(x\) of the right-hand side of equation (4.5), then \(h\geqslant 2\nu_0\), and hence in (4.6)
\[ l_1=l+\nu_0-h\leqslant l-\nu_0\leqslant l-(m_1!)\leqslant l-2. \]
We shall have the same two possibilities also for equations derived from (4.6). Therefore, if equation (4.1) admits a derived equation of series \(k\geqslant 1\)
\[ x^{l_k}\frac{dy_k}{dx}=E_k(x,y_k)A_{m_k}^{(k)}(x,y_k) \tag{4.7} \]
(where \(l_k\geqslant 1,\ m_k\geqslant 1,\ E_k\) is a function of type \(E\), and \(A_{m_k}^{(k)}(x,y_k)\) is a special polynomial in \(y_k\)), and if in (4.7) \(m_k\geqslant 2\), then we shall have \(1\leqslant l_k\leqslant l-2k\).
It follows from this that, for \(k>(l-1)/2\), the inequality \(m_k\geqslant 2\) cannot hold (otherwise \(l_k\leqslant l-2k<1\), which is impossible). Consequently, at least for every \(k\geqslant (l+1)/2\) in equation (4.7) we shall have \(m_k=1\), i.e., this equation will be a simple Briot and Bouquet equation. The theorem is proved.
Corollary. If the generalized Briot and Bouquet equation (4.1) has exponent \(m\geqslant 2\) and possesses the (!)-property, then the problem of the existence for it of \(O\)-curves with ordinary order of curvature \(\nu_0\) and with multiple measure of curvature \(\gamma_0\) is reduced, by means of substitutions of the form
\[ y=\gamma_0x^{\nu_0}+\gamma_1x^{\nu_0+\nu_1}+\cdots+\gamma_{k-1}x^{\nu_0+\nu_1+\cdots+\nu_{k-1}}+y_kx^{\nu_0+\nu_1+\cdots+\nu_{k-1}} \tag{4.8} \]
to the problem of the existence, for equations of the same type, of \(O\)-curves with zero, infinite, and special orders of curvature, and of \(O\)-curves with ordinary orders and with zero and infinite measures of curvature, and to the problem of the existence of \(O\)-curves for simple Briot and Bouquet equations (i.e., to problems whose solution may be regarded as known). In this connection it is sufficient to make no more than \(m\) distinct substitutions of the form (4.8), and in each of them \(1\leqslant k\leqslant (l+1)/2\).
This assertion follows from Theorems 3 and 5.
Let us now suppose that equation (4.1) has exponent \(m\geqslant 2\), but does not possess the (!)-property. Make in it the substitution \(x=\tilde{x}^{\,m!}\). We obtain an equation
\[ \tilde{x}^{\,\tilde l}\frac{dy}{d\tilde{x}}=\tilde E(\tilde{x},y)\tilde A_m(\tilde{x},y),\qquad \tilde l=m!(l-1)+1 \tag{4.9} \]
of the same form, possessing (by Lemma 1) the (!)-property with respect to the number \(m\), which is equivalent to the presence of the (!)-property in it. Theorem 5 is applicable to equation (4.9), and the role of the number \(n\) will be played by the number \(\tilde n\),
\[ \tilde n=\frac{\tilde l+1}{2}=\frac{m!(l-1)}{2}+1. \]
We formulate the result obtained in this way as a theorem.
Theorem 6. If the generalized Briot–Bouquet equation (4.1) has index \(m \geqslant 2\), but does not possess the \((!)\)-property, then the substitution \(x=\tilde{x}^{m!}\) transforms it into an equation of the same form (4.9), possessing the \((!)\)-property. If equation (4.9) admits derived equations up to the series \(\tilde n\) inclusive, then any one of them will be a simple Briot–Bouquet equation. If \(\{y=\varphi(\tilde{x})\}\) is the totality of all \(O\)-curves of equation (4.9), then \(\{y=\varphi(\sqrt[m!]{x})\}\) is the totality of all \(O\)-curves\(^1\) of equation (4.1).
It follows from Theorems 3 and 6 that, with the aid of no more than
\[ \tilde n=\frac{m!(l-1)}{2}+1 \]
operations, one can find all \(O\)-curves \(y=\varphi(\tilde{x})\) of the generalized Briot–Bouquet equation (4.9) and, consequently, all \(O\)-curves\(^1\)
\[ y=\psi(x)=\varphi\left(x^{\frac{1}{m!}}\right) \]
of equation (4.1). Here each operation consists of no more than \(m\) substitutions of the form
\[ y_{i-1}=\left(\gamma_{i-1}^{(j)}+y_i\right)x^{\nu_{i-1}^{(j)}}, \]
where \(i\) is the number of the operation, and \(j\) is the number of the substitution in the given operation.
Remark. Theorem 6 makes no claim to originality. If it is proved only for the special case \(l=1\), which is easy to do, then in the general case it follows from Theorem 3 of [3], supplemented by the particular result just mentioned. I note that in the proof of Theorem 3 of [3] there are inaccuracies. Theorem 6 may be regarded as a certain strengthening of Theorem 3 of [3] in its formulation and as a clarification of it in the sphere of the proof.
§ 5. Let us now suppose that in equation (1.1) \(n \geqslant 1\). Suppose, moreover, that it possesses the \((!)\)-property. Only the following three possibilities can occur:
1) Equation (1.1) has no ordinary (possible) order of curvature \(\nu\) to which a finite measure of curvature \(\gamma\) would correspond.
2) Equation (1.1) has such orders, but every finite measure of curvature \(\gamma\) corresponding to any such order is ordinary (in the corresponding equation of the form (2.7), \(P_1(\gamma)=0,\ P_2(\gamma)\ne 0\)).
3) Equation (1.1) has at least one ordinary (possible) order of curvature \(\nu\) to which a singular (finite) measure of curvature \(\gamma\) corresponds (in the corresponding equation of the form (2.7), \(P_1(\gamma)=0,\ P_2(\gamma)=0\)).
In the first case (as follows from the results of [2]) the problem of the existence, for equation (1.1), of \(O\)-curves with all possible orders of curvature is solved directly from the form of equation (1.1). Equation (1.1) has no derived equations in this case.
In the second case the same \(O\)-curves as in the first are possible and, in addition, \(O\)-curves of the form
\[ y=\gamma x^\nu+o(x^\nu),\qquad \nu>0\ \text{and}\ \gamma\ne 0\ \text{constants}. \tag{5.1} \]
The question of their actual presence (for fixed \(\nu\) and \(\gamma\)) is reduced, by means of the substitution
\(^1\) We are speaking of \(O^+\)-curves (in accordance with the convention adopted in the introduction).
\[ y=(\gamma+y_1)x^\nu \tag{5.2} \]
to the question of the existence of \(O\)-curves for the derived (from (1.1)) equation, which (by virtue of the presence in (1.1) of the (!)-property) has the form (3.1), and in this equation \(n_1=0\) (since \(\gamma\) is an ordinary measure of curvature). Consequently, it is a generalized Briot–Bouquet equation, and the behavior of its integral curves in a neighborhood of \((0,0)\) may be regarded as known (see § 4).
In the third case the equation (1.1) may have the same \(O\)-curves as in the second case, and, in addition to them, \(O\)-curves of the form (5.1), where \(\nu\) is an ordinary (possible) order of curvature and \(\gamma\) is the corresponding special measure of curvature. The problem of the existence of \(O\)-curves of this last form for fixed \(\nu\) and \(\gamma\) is reduced, by means of the substitution (5.2), to the problem of the existence of \(O\)-curves for the derived (from (1.1)) equation of the form (3.1), in which \(n_1\ge 1\) (since \(\gamma\) is a special measure of curvature). Consequently, this equation has the form of the original equation (1.1) and, moreover (according to Theorem 2), possesses the (!)-property. For it, in turn, the same three possibilities may present themselves:
1) the problem of the existence of \(O\)-curves is solved from the form of the equation itself; there are no derived equations;
2) the problem of the existence of \(O\)-curves is solved completely: a) partly from the form of the equation itself, b) partly by investigation of derived equations, each of which is a generalized Briot–Bouquet equation;
3) the problem of the existence of \(O\)-curves of a certain class leads to an analogous problem for derived equations having the same form as the original equation.
We shall show that this process cannot continue indefinitely; namely, we shall show that for derived (from (1.1)) equations of series \(k\), with \(k\) sufficiently large, the third possibility is not realized, and we shall find an upper estimate for this number \(k\).
§ 6. Preserving for equation (1.1) the assumption \(n\ge 1\), suppose that it possesses the complete (!)-property. Then it also possesses the (!)-property, and therefore everything said in § 5 remains valid for it. Moreover, Lemma 5 is applicable to it in this case, and, consequently, it is representable in the form (1.10) or (1.11). Define the number \(N\) by the formula
\[ N=\max\{r-mn+1;\ r-(m+1)(n-1)-l\}, \]
where the number \(r\) is defined by formula (1.6). It is obvious that \(N\ge 1\) is an integer.
Theorem 7. If equation (1.1) has exponent \(n\ge 1\), possesses the complete (!)-property, and has derived equations up to and including the \(N\)-th series, then any of its derived equations of series \(N\) will be a generalized Briot–Bouquet equation.
Proof. First case: \(m\ge 1\). In this case equation (1.1) is representable in the form (1.11). According to Corollary 2 of Theorem 2, any derived (from (1.1)) equation of any series \(k\), \(k=1,2,\ldots,N\), has the form (3.2). The presence for equation (1.1) of a fixed derived equation (3.2) of a given series \(k\) means that equation (1.1) admits a sequence (chain) of derived equations of the 0-th, 1st, \(\ldots\), \(k\)-th series (the derived equation of the 0-th series is the equation (1.1) itself), such that the equation of the \(i\)-th series (from this chain of equations): 1) has ordinary (possible) order of curvature \(\nu_i\), to which corresponds a finite (possible) measure of curvature \(\gamma_i\), \(i=0,1,2,\ldots,k-1\), 2) is obtained from the preceding equation of this sequence by means of the substitution
\[ y_{i-1}=(\gamma_{i-1}+y_i)x^{\nu_{i-1}},\qquad i=1,2,\ldots,k. \]
This, in turn, means that equation (3.2) can be obtained from equation (1.1) (or, what is the same thing, from equation (1.11)) by means of the substitution
\[ y=\gamma_0 x^{\nu_0}+\gamma_1 x^{\nu_0+\nu_1}+\ldots+\gamma_{k-1}x^{\nu_0+\nu_1+\ldots+\nu_{k-1}} +y_k x^{\nu_0+\nu_1+\ldots+\nu_{k-1}}, \]
which can be rewritten in the form
\[ y=\gamma_0 x^{\lambda_1}+\gamma_1 x^{\lambda_2}+\ldots+\gamma_{k-1}x^{\lambda_k}+y_k x^{\lambda_k} \equiv S_k(x)+y_k x^{\lambda_k}, \tag{6.1} \]
where \(\lambda_i=\nu_0+\nu_1+\ldots+\nu_{i-1}\) are positive integers, \(\lambda_i\geq i,\ i=1,2,\ldots,k\),
\[ S_k(x)\equiv \sum_{i=1}^{k}\gamma_{i-1}x^{\lambda_i} \]
is a polynomial in \(x\) of degree \(\lambda_k\). Consequently, equation (3.2) can be written in the form
\[ \begin{aligned} x^{\lambda_k+l}\frac{dy_k}{dx} &= \left[ E(x,y)\prod_{i=1}^{m}\bigl(x^{\lambda_k}y_k+S_k(x)-\varphi_i(x)\bigr) - x^l\bigl(S_k'(x)+\lambda_k x^{\lambda_k-1}y_k\bigr)\right.\\ &\qquad\left.\times \prod_{j=1}^{n}\bigl(x^{\lambda_k}y_k+S_k(x)-\psi_j(x)\bigr) \right] \left[ \prod_{j=1}^{n}\bigl(x^{\lambda_k}y_k+S_k(x)-\psi_j(x)\bigr) \right]^{-1}. \end{aligned} \tag{6.2} \]
Only two possibilities can present themselves:
1) there exists \(j_0\) such that
\[ \psi_{j_0}(x)\equiv S_k(x)+o(x^{\lambda_k}), \]
2) for every \(j,\ j=1,2,\ldots,n,\)
\[ S_k(x)-\psi_j(x)=O(x^{\sigma_j}), \]
\[ 1\leq \sigma_j\leq \lambda_k \]
(the symbol \(O(x^{\sigma_j})\) denotes an infinitesimal of order \(\sigma_j\) with respect to \(x\) as \(x\to 0\)).
It is obvious that the first of these possibilities cannot occur for arbitrarily large \(k\). We shall show that it does not occur for \(k>N\).
Let \(k\) be so large that
\[ \lambda_k>r-mn+1. \tag{6.3} \]
Since the numbers \(a_{ij}\) and \(l_{ij}\) in (1.7) are determined by the terms of the functions \(\varphi_i(x)\) and \(\psi_j(x)\) whose order with respect to \(x\) is not higher than \(r-mn+1\) (we shall call such terms of the functions \(\varphi_i(x)\) and \(\psi_j(x)\) principal), and
\[ S_k(x)-\psi_{j_0}(x)=o(x^{\lambda_k}), \]
then, under condition (6.3), all principal terms of the function \(\psi_{j_0}(x)\) enter into the polynomial \(S_k(x)\). Therefore
\[ \varphi_i(x)-S_k(x)\equiv a_{ij_0}x^{l_{ij_0}}+o(x^{l_{ij_0}}),\qquad i=1,2,\ldots,m, \]
\[ \prod_{i=1}^{m}\bigl(x^{\lambda_k}y_k+S_k(x)-\varphi_i(x)\bigr) = x^h \prod_{i=1}^{m}\bigl(x^{\lambda_k-l_{ij_0}}y_k-a_{ij_0}+o(1)\bigr), \]
where
\[ h=\sum_{i=1}^{m} l_{ij_0}=r-\sum_{i;\,j\ne j_0} l_{ij}\le r-m(n-1),\qquad \lambda_k-l_{ij_0}\ge 0,\quad i=1,2,\ldots,m. \]
As for the product
\[ \prod_{j=1}^{n}\bigl(x^{\lambda_k}y_k+S_k(x)-\psi_j(x)\bigr), \tag{6.4} \]
it contains the factor
\[ x^{\lambda_k}y_k+S_k(x)-\psi_{j_0}(x)=x^{\lambda_k}y_k+o(x^{\lambda_k}) \]
and therefore has, with respect to \(x\), order \(h_k\ge \lambda_k+n-1\). Thus, the first term in the numerator of the right-hand side of equation (6.2) is representable in the form \(x^h\Phi_k(x,y_k)\), where \(h\le r-m(n-1)\), \(\Phi_k(0,0)\ne0\), while the second is representable in the form \(x^{g_k}\Psi_k(x,y_k)\), where
\[ g_k=l+\lambda_1-1+h_k\ge l+n+\lambda_1+\lambda_k-2,\qquad \Psi_k(0,0)=0. \]
Therefore, for \(h\le g_k\), which occurs if
\[ \lambda_k\ge r-(m+1)(n-1)-l-(\lambda_1-1), \tag{6.5} \]
equation (6.2) (or, what is the same, equation (3.2)) takes the form
\[ x^{l_k}\frac{dy_k}{dx} = \frac{E_k(x,y_k)}{B_{n_k}^{(k)}(x,y_k)}, \tag{6.6} \]
which contradicts the fact that in equation (3.2) \(m_k\ge1\) (i.e., the fact that the equation of the \((k-1)\)-st series, “primordial” for (3.2), has ordinary curvature order \(\nu_{k-1}\) and the corresponding finite curvature measure \(\gamma_{k-1}\)).
Consequently, if \(k\) is so large that \(\lambda_k\) satisfies inequalities (6.3) and (6.5), then the identity
\[ \psi_j(x)\equiv S_k(x)+o(x^{\lambda_k}) \]
is impossible for any \(j\), \(j=1,2,\ldots,n\). And since \(\lambda_1=\nu_0\) and, for any \(k\ge1\), \(\lambda_k\ge k\), inequalities (6.3) and (6.5) will be fulfilled for
\[ k\ge N_1=\max\{\,r-mn+1,\; r-(m+1)(n-1)-l-(\nu_0-1)\,\} \tag{6.7} \]
and still more so for \(k>N\).
Suppose now that, for some \(k\ge1\), the second of the two possibilities indicated above is realized (from the preceding considerations it follows that this will necessarily occur for \(k\le N\)). Then the product (6.4) can be represented in the form
\[ x^{\sigma_0}\prod_{j=1}^{n}\bigl(x^{\lambda_k-\sigma_j}y_k+c_j+o(1)\bigr) \]
(where \(\sigma_0=\sum_{j=1}^n \sigma_j,\ \lambda_k-\sigma_j>0,\ c_j\ne 0\) are constants, \(j=1,2,\ldots,n\)), i.e. in the form
\[ x^{\sigma_0}\Psi_k(x,y_k),\qquad \sigma_0>0,\qquad \Psi_k(0,0)\ne 0. \]
Therefore, rewriting equation (6.2) in the form (3.2), we obtain
\[ x^{l_k}\frac{dy_k}{dx}=E_k(x,y_k)A_{m_k}^{(k)}(x,y_k), \]
and this is a generalized Briot–Bouquet equation.
The 2nd case: \(m=0\) (and, consequently, \(r=0\) (see § 1)). In this case equation (1.1) is representable in the form (1.10). Substitution (6.1) transforms it to the form
\[ \begin{aligned} x^{\lambda_k+l}\frac{dy_k}{dx} &= \left[ E(x,S_k(x)+y_kx^{\lambda_k}) - x^l\left(S'_k(x)+\lambda_k x^{\lambda_k-1}y_k\right) \prod_{j=1}^n\left(x^{\lambda_k}y_k+S_k-\psi_j\right) \right] \\ &\quad{}\times \left[ \prod_{j=1}^n\left(x^{\lambda_k}y_k+S_k(x)-\psi_j(x)\right) \right]^{-1}. \end{aligned} \tag{6.8} \]
The same two possibilities may occur as in the 1st case. Let us consider the first of them. Suppose \(k>1\). Then also \(\lambda_k>1\), and therefore the first term on the right-hand side of (6.8) tends, as \(x=0\), to the number \(E(0,0)\ne 0\), while the second term is the same as in the 1st case. Therefore, for \(g_k>0\), i.e. for \(\lambda_k>1-l-n-(\lambda_1-1)\), equation (6.8) (or, what is the same, equation (3.2)) assumes the form (6.6), which is impossible (for the same reason as in the 1st case). Consequently, for
\[ k>\max\{1,\,1-n-l-(\nu_0-1)\}=N_1 \]
and, all the more, for
\[ k>\max\{1,\,1-n-l\}=N \]
the first possibility is not realized. The consideration of the second possibility in the case \(m=0\) does not differ from that for the case \(m>1\).
Thus, if equation (1.1) has derived equations of series \(1,2,\ldots,N\), then at least in the \(N\)-th series each of these equations will be a generalized Briot–Bouquet equation. The theorem is proved.
Corollary. If equation (1.1) possesses the full (!)-property, then the problem of whether it has \(O\)-curves with ordinary order of curvature \(\nu_0\) and special measure of curvature \(\gamma_0\) is reduced, by substitutions of the form
\[ y=\gamma_0x^{\nu_0}+\gamma_1x^{\nu_0+\nu_1}+\cdots+\gamma_{k-1}x^{\nu_0+\nu_1+\cdots+\nu_{k-1}} +y_kx^{\nu_0+\nu_1+\cdots+\nu_{k-1}} \tag{6.9} \]
to the problem of the existence, for equations of the same type, of \(O\)-curves with zero, infinite, and special orders of curvature and of \(O\)-curves with ordinary orders and with zero, infinite, and ordinary measures of curvature. In this it is sufficient to make no more than \(L\) different substitutions of the form (6.9), and in each of them \(1\le k\le N\).
This assertion follows from Theorems 3 and 5. In its formulation the number \(N\) may be replaced by the number \(N_1\), which is determined by formula (6.7).
§ 7. Let us now suppose that equation (1.1), in which still \(n>1\), does not possess the full (!)-property. In it make the substitution of the independent variable \(x\) according to the formula \(x=\tilde{x}^{M!}\) (the number \(M\) is determined by formula (3.3)). We obtain the equation
\[ \tilde{x}^{\tilde{l}}\frac{dy}{d\tilde{x}} = \tilde{E}(\tilde{x},y)\frac{\tilde{A}_m(\tilde{x},y)}{\tilde{B}_n(\tilde{x},y)} \tag{7.1} \]
\[
(\tilde{l}=M!(l-1)+1)
\]
of the same form, which, according to Lemma 1, possesses the (!)-property with respect to the number \(M\). In equation (7.1) the exponents \(m\) and \(n\) are the same as in equation (1.1), while the exponent \(\tilde{l}\) has the same sign as the exponent \(l\) in equation (1.1). Therefore, if \(\tilde{K}\) and \(\tilde{M}\) are numbers having for (7.1) the same meaning as the numbers \(K,M\) have for (1.1) (the number \(K\) is determined by formula (1.8)), then \(\tilde{K}=K,\ \tilde{M}=M\). It follows from this that the presence in (7.1) of the (!)-property with respect to \(M\) is equivalent to the presence in it of the full (!)-property. But then Theorem 7 is applicable to equation (7.1).
Let \(\tilde{r}\) be the order with respect to \(\tilde{x}\) of the resultant of the special polynomials in \(y\) occurring in the numerator and denominator of the right-hand side of (7.1). Obviously, \(\tilde{r}=M!r\). Let
\[ \tilde{N}=\max\{\tilde{r}-mn+1,\quad \tilde{r}-(m+1)(n-1)-\tilde{l}\}= \]
\[ =\max\{M!r-mn+1,\quad M!(r-l+1)-(m+1)(n-1)-1\}. \tag{7.2} \]
Applying Theorem 7 to (7.1), we obtain the following result for equation (1.1).
Theorem 8. If equation (1.1), in which \(n>1\), does not possess the full (!)-property, then the substitution \(x=\tilde{x}^{M!}\) (the number \(M\) is determined by formula (3.3)) transforms it into an equation of the same form (7.1), possessing the full (!)-property. If equation (7.1) admits derived equations up to order \(\tilde{N}\) inclusive, then any of them will be a generalized Briot and Bouquet equation. If \(\{y=\varphi(\tilde{x})\}\) is the totality of all \(O\)-curves of equation (7.1), then \(\{y=\varphi(\sqrt[M!]{x})\}\) is the totality of all \(O\)-curves¹ of equation (1.1).
§ 8. We now give an example contradicting Theorem 4 of the article [3], which was mentioned in the introduction.
Suppose that equation (1.1) possesses the (!)-property with respect to the number \(K\) (see (1.8)). The theorem just mentioned is formulated as follows (I change only the notation and indices):
“If the origin is an isolated singular point for equation (1.1), then, by means of no more than \(L\) substitutions of the form
\[ y=\gamma_{1,i}x^{\lambda_{1,i}}+\gamma_{2,i}x^{\lambda_{2,i}}+\cdots+\gamma_{k,i}x^{\lambda_{k,i}}+u_i x^{\lambda_{k,i}} \tag{8.1} \]
one can obtain all generalized Briot and Bouquet equations corresponding to the various orders and measures of curvature of the characteristics of equation (1.1). Here the number \(k\) in each of the substitutions (8.1) is determined by the inequality \(k\le mr\), where \(r\) is the lowest term of the analytic function (1.5), which under the conditions of our theorem must not be identically equal to zero.”
¹ The discussion concerns \(O^+\)-curves.
The estimate \(k \le mr\) delivered by this theorem is incorrect, as is easily seen from the following example.
Example
\[ y'=-\frac{x^{3}(y-x)}{y-x^{2}} . \tag{8.2} \]
This is an equation of the form (1.1), and here \(l=-3,\ m=1,\ n=1,\ K=1,\ L=2,\ M=2;\ R(x)\equiv x-x^{2},\ r=1,\ mr=1\). It has the (!)-property with respect to \(K\). The theorem of I. S. Kukles and D. M. Gruz asserts: with the aid of no more than two substitutions of the form (8.1), one can obtain all generalized Briot and Bouquet equations corresponding to the various orders and measures of curvature of the characteristics of equation (8.2), and in each of these substitutions \(k=1\).
In reality the situation is otherwise. Investigating equation (8.2) by Frommer’s method, we easily establish the following.
The orders of curvature of its \(O\)-curves can only be the numbers 2 and 3.
For curves with order of curvature 2, the only possible measure is the number 1. Introducing into (8.2) the substitution \(y=(1+y_1)x^2\), we obtain the derived equation of the 1st series
\[ xy_1'=\frac{-x-2y_1+x^2-2y_1^2+x^2y_1}{y_1}, \tag{8.3_1} \]
which is not a generalized Briot and Bouquet equation. For the \(O\)-curves of equation \((8.3_1)\), the only possible order of curvature is the number 1, and the only possible measure of curvature is the number \(-\dfrac{1}{2}\). Introducing into \((8.3_1)\) the substitution \(y_1=\left(-\dfrac{1}{2}+y_2\right)x\), we obtain the equation of the 2nd series derived (from (8.2)),
\[ x^2y_2'=-\frac{1}{2}x+4y_2+\text{terms of higher orders}, \tag{8.4} \]
which is a simple Briot and Bouquet equation.
For order of curvature 3, the only possible measure of curvature is the number \(\dfrac{1}{3}\). Transforming (8.2) by means of the substitution \(y=\left(\dfrac{1}{3}+y_1\right)x^3\), we obtain for it the second derived equation of the 1st series,
\[ xy_1'=\frac{1}{3}x-3y_1+\text{terms of higher orders}. \tag{8.3_2} \]
It is a simple Briot and Bouquet equation.
Thus, the investigation of the \(O\)-curves of equation (8.2) is indeed reduced, by means of two substitutions of the form (8.1),
\[ y=\left(\frac{1}{3}+y_1\right)x^3 \quad\text{and}\quad y=x^2-\frac{1}{2}x^3+y_2x^3 \]
to the investigation of the \(O\)-curves of the generalized Briot and Bouquet equations \((8.3_2)\) and (8.4); however, in one of these substitutions the number \(k=2\) (contrary to the assertion of the theorem of I. S. Kukles and D. M. Gruz).
Theorem 8 of the present paper requires a preliminary transformation of equation (8.2) by means of the substitution \(x=\hat{x}^{2}\), and for the resulting equation gives the estimate \(k\leq N=9\). This estimate is very much overstated (it is obvious that also for the transformed equation \(k\leq 2\)), but it has the merit of reliability.
References
- Frommer M. UMN, issue IX, 1941.
- Andreev A. F. Vestnik Leningrad. Univ., No. 1, 1962.
- Kukles I. S. and Gruz D. M. Izv. AN UzSSR, series of physical and mathematical sciences, No. 1, 1958.
- Markushevich A. I. Theory of Analytic Functions. Moscow, GITTL, 1950.
- Kurosh A. G. Higher Algebra. Gostekhizdat, 1946.
- Khaimov N. B. Some theorems on singular points of the first group: Scientific Notes of the Stalinabad Pedagogical Institute, vol. 1, 1952.
- Gruz D. M. Proceedings of the Uzbek University, issue 78, 1958.
- Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations. Moscow, GITTL, 1949.
- Andreev A. F. On Frommer’s distinction problems. Proceedings of the Fourth All-Union Mathematical Congress, vol. II. Leningrad, “Nauka,” 1964.
- Khaimov N. B. Scientific Notes of the Stalinabad Pedagogical Institute, vol. II. Mathematics, Methodology. 1952.
Received by the editors
April 23, 1965
Leningrad Institute
of Precision Mechanics and Optics