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UDC 517.916
ON A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH ALGEBRAIC MOVING SINGULAR POINTS
B. P. Bogoslovskii
Introduction. Painlevé investigated equations of the form (see [1])
\[ y''=R(x,y,y'), \tag{0.1} \]
where \(R\) is a rational function of \(y, y'\) and analytic in \(x\). His method consisted in isolating the necessary conditions under which equation (0.1) contains no movable critical singular points. He also found sufficient conditions that made it possible to distinguish classes of equations having only single-valued movable singular points of pole type.
N. P. Erugin [2, 3] considered systems of a more general form:
\[ \frac{dx}{dz}=f_1(x,y,z),\qquad \frac{dy}{dz}=f_2(x,y,z). \tag{0.2} \]
He indicated sufficient conditions that the functions \(f_1\) and \(f_2\) must satisfy in order that system (0.2) have no movable singular points of essentially singular type. In this way systems were singled out that contain, along with single-valued singular points, also multivalued singular points. In [2] a method was also proposed for constructing a solution in a neighborhood of a movable singular point in the form of asymptotic series.
In the present article equations of the form
\[ x''=a_0x^k+f(x,z), \tag{0.3} \]
are considered, where \(a_0\) is a constant, \(k\) is a natural number, and \(f(x,z)\) is a polynomial in \(x,z\), with the exponent of \(x\) in the leading term of the polynomial being \(k_1<k\). The article constructs asymptotic series, finds conditions under which these series are convergent, carries out the inversion of convergent series, and distinguishes classes of equations of the form (0.3) with algebraic movable singular points.
Replace equation (0.3) by the system
\[ \frac{dx}{dz}=y,\qquad \frac{dy}{dz}=a_0x^k+f(x,z). \tag{0.4} \]
As is known [2], every solution of system (0.4) in a neighborhood of a movable singular point \(z_0\) has the property \(x\to\infty,\ y\to\infty\) as \(z\to z_0\). It is easy to see that
\[ y=Ax^{\frac{k+1}{2}}\left[1+o\left(x^{k_1-k}\right)\right], \tag{0.5} \]
where \(A\) is determined by the exponent \(k\) and the coefficient \(a_0\).
writing the system (0.4) in the form
\[ dz=\frac{dx}{y},\qquad ydy=\left[a_0x^k+f(x,z)\right]dx, \tag{0.6} \]
we shall construct asymptotic series by the method of iterations. Successively substituting (0.5) into the first equality (0.6), and the value of \(z\) thus found into the second equality (0.6), we obtain a new expansion for \(y\):
\[ y=Ax^{\frac{k+1}{2}}\left[1+a_1x^{-\beta_1}+a_2x^{-\beta_2}+\cdots+a_jx^{-\beta_j}+o(x^{-\beta})\right], \tag{0.7} \]
where \(a_j\) are uniquely determined through the coefficients of the polynomial \(f(x,z)\) and \(A\), and \(k_1-k=-\beta_1>-\beta_2>-\beta_3>\cdots>-\beta_j>-\beta\).
It is clear that such a process can be continued indefinitely. What is essential is that at each new iteration the coefficients of the previously determined terms do not change, while the coefficients obtained as a result of carrying out the given iteration are uniquely determined through the coefficients of the polynomial \(f(x,z)\) and those already found. Since the passage from equality (0.5) to (0.7) also includes integration, \(\ln x\) may appear in the expansion (0.7), but the coefficients of the equation can always be chosen so that the asymptotic expansion will not contain \(\ln x\).
§ 1. CONSTRUCTION OF ASYMPTOTIC SERIES
Without loss of generality, we shall consider equation (0.3) in the form
\[ x''=\frac{k+1}{2}x^k+\sum_{j=1}^{k}\sum_{i=0}^{p}a_{ij}z^ix^{k-j}. \tag{1.1} \]
Put
\[ y=Ax^{\frac{k+1}{2}}\left(1+\sum_{n=1}^{\infty}A_nx^{-\frac{n}{2}}\right), \tag{1.2} \]
where the coefficients \(A_n\) and \(A\) are to be determined. Substituting (1.2) into the first equality (0.6), after integration we obtain
\[ z-z_0=-\frac{1}{A}x^{-\frac{k-1}{2}}\sum_{n=0}^{\infty}L_nx^{-\frac{n}{2}}, \tag{1.3} \]
where
\[ L_0=\frac{2}{k-1};\qquad L_n=\frac{2}{k+n-1}D_n \tag{1.4} \]
and
\[ D_n=-A_n-\sum_{i+j=n}A_iD_j. \tag{1.5} \]
We substitute the found value of \(z\) into the second equality (0.6). After transformations we obtain
\[ ydy=\left(\frac{k+1}{2}x^k+\sum_{n=1}^{\infty}R_nx^{\frac{2k-n}{2}}\right)dx. \tag{1.6} \]
Here
\[ R_n=\sum_{i=0}^{p}\sum_{s=0}^{i}\frac{(-1)^s}{A^s}C_i^s z_0^{\,i-s} \sum_{2j+r=n} a_{ij}L_{r-s(k-1)}^{(s)}, \tag{1.7} \]
\[ L_0^{(0)}=1,\quad L_n^{(0)}=0,\quad L_n^{(s)}=\sum \frac{s!}{\alpha_1!\alpha_2!\ldots\alpha_k!} L_{n_1}^{\alpha_1}L_{n_2}^{\alpha_2}\ldots L_{n_k}^{\alpha_k}, \tag{1.8} \]
where \(\alpha_1+\alpha_2+\cdots+\alpha_k=s\) and \(n_1\alpha_1+n_2\alpha_2+\cdots+n_k\alpha_k=n\).
Since in equality (1.6), for \(n=2k+2\), there occurs a term containing \(x^{-1}\), integration will produce \(\ln x\). It is clear that, putting
\[ R_{2k+2}=0, \tag{1.9} \]
we obtain a condition under which \(\ln x\) will not be contained in the asymptotic expansions (1.2) and (1.3).
Integrating (1.6) under condition (1.9), we shall have
\[ y^2=x^{k+1}\left(1+2\sum_{n=1}^{\infty}T_n x^{-\frac n2}\right), \tag{1.10} \]
where
\[ T_n=\frac{2}{2k+2-n}R_n,\quad T_{2k+2}=C=\mathrm{const}. \tag{1.11} \]
Extracting the square root from (1.10), with
\[ A=\pm 1,\quad A_n=T_n-\frac12\sum_{i+j=n}A_iA_j, \tag{1.12} \]
we obtain (1.2).
Thus, the coefficients of the asymptotic expansion for \(y\) in (1.2) are completely determined by the recurrence formulas (1.12), (1.11), (1.8), (1.7), (1.5), and (1.4).
Since \(L_n\) are expressed in terms of \(A_n\), the coefficients of the asymptotic expansion of \(z-z_0\) in (1.3) have also been found. It is easy to show that for odd \(k=2l+1\), the expansions (1.2) and (1.3) will contain no terms with fractional powers of \(x\).
§ 2. PROOF OF THE CONVERGENCE OF THE ASYMPTOTIC SERIES AND THEIR INVERSION
We shall prove the convergence of the series by constructing a new system having a holomorphic solution. We shall again follow the method of proof proposed by N. P. Erugin in [2]. Introducing the variables
\[ u=\frac1A yx^{-\frac{k+1}{2}}-1,\quad v=-A(z-z_0)x^{\frac{k-1}{2}}-\frac{2}{k-1},\quad t=x^{-\frac12}, \tag{2.1} \]
where the constant number \(A\) will be chosen below, after transformations we obtain
\[ t\frac{du}{dt}=(k+1)(u+1)-\frac{1}{A^2}\left\{k+1+2\sum_{j=1}^{k}\sum_{i=0}^{p}a_{i,j}\left[z_0-\right.\right. \]
\[
-\frac{1}{A}t^{k-1}\left(\frac{2}{k-1}+v\right)^i
\]^{t^{2j}}
\left[1+\sum_{l=1}^{\infty}(-1)^l u^l\right],
]
\[ t\frac{dv}{dt}=2\sum_{l=1}^{\infty}(-1)^l u^l-(k-1)v. \tag{2.2} \]
Choose \(A\) so that in the first equation of system (2.2) the constant term becomes zero. This will occur if
\[
k+1-\frac{k+1}{A^2}=0,
\]
i.e. \(A=\pm 1\), which agrees with (1.12). Separating out the linear terms, we rewrite system (2.2) in the form
\[ t\frac{du}{dt}=(2k+2)u+F_1(u,v,t), \]
\[ t\frac{dv}{dt}=-2u-(k-1)v+F_2(u). \tag{2.3} \]
Here the functions \(F_1\) and \(F_2\) contain neither constant terms nor first powers of \(u\) and \(v\). Since the roots of the equation
\[ \begin{vmatrix} (2k+2)-\lambda & 0\\ -2 & -(k-1)-\lambda \end{vmatrix}=0, \]
\(\lambda_1=2k+2,\ \lambda_2=-(k-1)\), are integers of different signs, on the basis of [5] we conclude that system (2.3) has a family of solutions possessing the property \(u\to 0,\ v\to 0\) as \(t\to 0\) and containing one arbitrary constant.
We shall seek the solution of (2.2) in the form
\[ u=\sum_{n=1}^{\infty}u_n t^n,\qquad v=\sum_{n=1}^{\infty}v_n t^n. \tag{2.4} \]
Substituting (2.4) into (2.2), we find recurrence formulas for determining \(u_n\) and \(v_n\):
\[ 2(2k-n+2)u_n=4R_n-(2k-n+2)\sum_{i+j=n}u_i u_j, \]
\[ 2u_n+(k+n-1)v_n=-2\sum_{i+j=n}u_iD_j, \tag{2.5} \]
where
\[ D_n=-u_n-\sum_{i+j=n}u_iD_j, \tag{2.6} \]
\[ R_n=\sum_{i=0}^{p}\sum_{s=0}^{i}\frac{(-1)^s}{A^s}C_i^s z_0^{\,i-s} \sum_{2j+r=n}a_{ij}v_{r-s(k-1)}^{(s)}, \tag{2.7} \]
\[ v_0^{(0)}=1,\qquad v_n^{(0)}=0,\qquad v_n^{(s)}=\sum \frac{s!}{\alpha_1!\alpha_2!\cdots\alpha_k!}\, v_{n_1}^{\alpha_1}v_{n_2}^{\alpha_2}\cdots v_{n_k}^{\alpha_k}. \tag{2.8} \]
It is clear from this that, for the existence of a holomorphic solution of system (2.2), it is necessary that \(R_{2k+2}=0\). In this case the coefficient \(u_{2k+2}\) remains arbitrary. The condition obtained is also sufficient, as follows from [5].
Putting
\[ u_{2k+2}=C-\frac{1}{2}\sum_{i+j=2k+2}u_i u_j, \tag{2.9} \]
where \(C\) is the same constant as in (1.11), we verify the equality of all \(A_n=u_n\) and \(L_n=v_n\), and thereby prove the convergence of the series
\[ \sum_{n=1}^{\infty} A_n x^{-\frac{n}{2}},\qquad \sum_{n=1}^{\infty} L_n x^{-\frac{n}{2}} \tag{2.10} \]
in powers of \(x^{-\frac{1}{2}}\).
It is easy to see that the inversions (1.2) and (1.3) lead to the expansions:
1) for \(k=2l\)
\[ x=\sum_{n=-2}^{\infty} S_n (z-z_0)^{\frac{n}{2l-1}},\qquad y=\sum_{n=-(2l+1)}^{\infty} T_n (z-z_0)^{\frac{n}{2l-1}}; \tag{2.11} \]
2) for \(k=2l+1\)
\[ x=\sum_{n=-1}^{\infty} S_n (z-z_0)^{\frac{n}{l}},\qquad y=\sum_{n=-(l+1)}^{\infty} T_n (z-z_0)^{\frac{n}{l}}, \tag{2.12} \]
and all the series converge in a neighborhood of \(z_0\).
Hence it is clear that the movable singular point \(z_0\) is algebraic. In the first case the equation has a one-parameter family of solutions, and in the second—two one-parameter families.
§ 3. EQUATIONS WITH ALGEBRAIC MOVABLE SINGULAR POINTS
It is easy to show that condition (1.9), for \(k=2l\), reduces to an algebraic equation in \(z_0\) of degree \(2p-2\). If this equation is written in the form
\[ \sum_{i=0}^{2p-2} R_i z_0^i=0, \tag{3.1} \]
then the coefficients \(R_i\) are found from the formulas: 1) for \(l=1\)
\[ R_i=\sum_{r+s=i}\bigl[(r+1)(s+1)a_{r+1,1}a_{s+1,1} +(r+1)(r+2)a_{r+2,1}a_{s,1}\bigr]- \]
\[ -(i+1)(i+2)(i+3)(i+4)a_{i+4,1} -3(i+1)(i+2)a_{i+2,2}; \tag{3.2} \]
2) for \(l>1\)
\[ R_i=\frac{2l-1}{l(2l+1)} \sum_{r+s=i}\bigl[(r+1)(s+1)a_{r+1,1}a_{s+1,1} +(r+1)(r+2)a_{r+2,1}a_{s,1}\bigr]- \]
\[ -(i+1)(i+2)a_{i+2,2}, \tag{3.3} \]
where \(i=0,1,2,\ldots,2p-2\), and everywhere it is necessary to retain only those \(a_{i,j}\) whose first index is not greater than \(p\).
Condition (3.1) makes it possible to distinguish the following classes of equations of the form (1.1) for \(k=2l\).
-
All \(R_i=0\) \((i\ne 0)\), \(R_0\ne 0\). The equation has not a single algebraic movable singular point in a neighborhood of which the solution can be constructed in the form (2.11).
-
There exists \(R_i\ne 0\) \((i\ne 0)\). The equation has no more than \(2p-2\) algebraic movable singular points.
-
All \(R_i=0\). In a neighborhood of each movable singular point the solution can be constructed in the form (2.11).
It is clear that any equation of the form (1.1) for \(k=2l\) with fixed coefficients \(a_{i,j}\) falls into one of the classes, and all the classes are nonempty. The following holds.
Theorem 1. There exist no other algebraic movable singular points of equation (1.1) for \(k=2l\), except those singled out in classes 2, 3.
Proof. Let the algebraic singular point \(z_0^*\) be such that in its neighborhood there is an expansion
\[ x=\sum_{n=-t}^{\infty} S_n (z-z_0^*)^{\frac{n}{r}} . \tag{3.4} \]
Substituting (3.4) into equation (1.1), we find \(t=2\), \(r=2l-1\), which coincides with (2.11). The recurrence formulas for determining the coefficients \(S_n\) have the form
\[ -\frac{(m+4l-2)(m+2l-1)}{(2l-1)^2}\,S_{m+4l-2} = \]
\[ =\frac{2l+1}{2}S_m^{(2l)} +\sum_{j=1}^{2l}\sum_{i=0}^{p} \sum_{(2l-1)j+r=m} b_{i,j} S_r^{(2l-j)} . \tag{3.5} \]
Here
\[ b_{0,j}=\sum_{i=0}^{p} a_{i,j} z_0^{*\,i}, \qquad b_{i,j}=\frac{1}{i!}\frac{d^i}{dz^i}\left(\sum_{i=0}^{p} a_{i,j} z^i\right)\Bigg|_{z=z_0^*}, \tag{3.6} \]
\[ S_0^{(0)}=1,\qquad S_m^{(0)}=0,\qquad S_m^{(s)}=\sum \frac{s!}{\alpha_1!\alpha_2!\ldots \alpha_k!}\, S_{n_1}^{\alpha_1} S_{n_2}^{\alpha_2}\ldots S_{n_k}^{\alpha_k}, \tag{3.7} \]
where \(\alpha_1+\alpha_2+\cdots+\alpha_k=s\), \(n_1\alpha_1+n_2\alpha_2+\cdots+n_k\alpha_k=m\), and the smallest value that the \(n_i\) can take is \(-2\).
It is easy to show that the coefficient \(S_{4l}\) must be arbitrary; consequently, for \(m=2\) in equality (3.5) we have an identity. This identity is the condition under which, in a neighborhood of the point \(z_0^*\), the solution can be constructed in the form (3.4). It can be shown that the condition obtained coincides with the previously found one (3.1), which proves the theorem.
Condition (1.9) for \(k=2l+1\) reduces to the equation
\[ \sum_{i=0}^{p(l+2)-1} P_i^{(\lambda)} z_0^i=0 . \tag{3.8} \]
- If \(l=1\), then
\[ \begin{aligned} P_i^{(A)}={}& \sum_{r+s+t=i} (r+1)a_{r+1,1}a_{s,1}a_{t,1} +18(i+1)a_{i+1,3} \\ &+3(i+1)(i+2)(i+3)a_{i+3,1} -3\sum_{r+s=i}(r+1)(a_{r+1,1}a_{s,2}+a_{r+1,2}a_{s,1}) \\ &+3A\left\{\sum_{r+s=i}\left[(r+1)(s+1)a_{r+1,1}a_{s+1,1} +(r+1)(r+2)a_{r+2,1}a_{s,1}\right]\right. \\ &\left.\hspace{5.5em}-3(i+1)(i+2)a_{i+2,2}\right\}. \end{aligned} \tag{3.9} \]
- If \(l>1\), then
\[ \begin{aligned} P_i^{(A)}={}& \sum_{j+r=i+l+2}\sum_{s+t=i}\frac{s+1}{r+l}\,a_{s+1,j}\beta_t^{(r)} \\ &+A\left\{\frac{1}{l(l+1)(2l+1)} \sum_{r+s=i}\left[(r+1)(s+1)a_{r+1,1}a_{s+1,1}\right.\right. \\ &\left.\left.\hspace{7.5em}+(r+1)(r+2)a_{r+2,1}a_{s,1}\right] -\frac{(i+1)(i+2)}{2l^2}a_{i+2,2}\right\}, \end{aligned} \tag{3.10} \]
where \(i=0,1,2,\ldots,p(l+2)-1\). Here
\[ \beta_0^{(0)}=1,\qquad \beta_i^{(0)}=0,\qquad \beta_i^{(n)}=-\frac{1}{2l+2-n}\,a_{i,n} +\sum_{k+j=n}\sum_{r+s=i}\alpha_r^{(k)} \left[\frac{1}{2}\alpha_s^{(j)}-\beta_s^{(j)}\right]; \tag{3.11} \]
\[ \alpha_i^{(n)}=\frac{1}{2l+2-n}\,a_{i,n} -\frac{1}{2}\sum_{k+j=n}\sum_{r+s=i}\alpha_r^{(k)}\alpha_s^{(j)}. \]
Since \(A=\pm1\), condition (3.8) splits into two equations with respect to \(z_0\), which makes it possible to distinguish the following classes of equations of the form (1.1) for \(k=2l+1\).
-
All \(P_i^{(A)}=0\) \((i\ne0)\), \(P_0^{(+1)}\ne0\), \(P_0^{(-1)}\ne0\). The equations have no algebraic movable singular points.
-
There is \(P_i^{(+1)}\ne0\) \((i\ne0)\) \((P_i^{(-1)}\ne0,\ i\ne0)\); all \(P_i^{(-1)}=0\) \((i\ne0)\), \(P_0^{(-1)}\ne0\) \((P_i^{(+1)}=0\ (i\ne0),\ P_0^{(+1)}\ne0)\). The equations have no more than \(p(l+2)-1\) algebraic movable singular points, in neighborhoods of which one can construct one-parameter families of solutions of the form (2.12) for
\[ S_{-1}=-\sqrt[l]{\frac{1}{l}} \qquad \left(S_{-1}=\sqrt[l]{\frac{1}{l}}\right). \]
- There are \(P_i^{(-1)}\ne0\) \((i\ne0)\), \(P_i^{(+1)}\ne0\) \((i\ne0)\). The equations have no more than \(p(l+2)-1\) algebraic movable singular points, in neighborhoods of which one can construct one-parameter families of solutions
of the form (2.12) for
\[ S_{-1}=\sqrt[l]{\frac{1}{l}}, \]
and no more than \(p(l+2)-1\) algebraic movable singular points, in neighborhoods of which one can also construct one-
parametric families of solutions of the form (2.12), but with \(S_{-1}=\sqrt[l]{\dfrac{1}{l}}\). (Among the distinguished movable singular points there may also be such points that in their neighborhood there exist two one-parameter families of solutions of the form (2.12) with \(S_{-1}=\pm \sqrt[l]{\dfrac{1}{l}}\).)
-
All \(P_i^{(+1)}=0\) \((P_i^{(-1)}=0)\), \(P_i^{(-1)}=0\), \(P_0^{(-1)}\ne 0\) \((P_i^{(+1)}=0,\ P_0^{(+1)}\ne 0)\). In the neighborhood of any movable singular point one can construct one family of solutions of the form (2.12) with
\[ S_{-1}=-\sqrt[l]{\frac{1}{l}} \qquad \left(S_{-1}=\sqrt[l]{\frac{1}{l}}\right). \] -
All \(P_i^{(+1)}=0\) \((P_i^{(-1)}=0)\). There is \(P_i^{(-1)}\ne 0,\ i\ne 0\) \((P_i^{(+1)}\ne 0,\ i\ne 0)\). In the neighborhood of any movable singular point one can construct one family of solutions of the form (2.12) with \(S_{-1}=-\sqrt[l]{\dfrac{1}{l}}\) \(\left(S_{-1}=\sqrt[l]{\dfrac{1}{l}}\right)\), but the equation has still other algebraic movable singular points, in whose neighborhoods the solution can be constructed in the same form, but taking
\[ S_{-1}=\sqrt[l]{\frac{1}{l}} \qquad \left(S_{-1}=-\sqrt[l]{\frac{1}{l}}\right). \]
The number of such points is not greater than \(i\), where \(i\) is the greatest exponent of \(z_0\) in equation (3.8) for \(A=1\) \((A=-1)\). -
All \(P_i^{(+1)}=0,\ P_i^{(-1)}=0\). In the neighborhood of each movable singular point one can construct two one-parameter families of solutions.
It is clear that any equation of the form (1.1) with \(k=2l+1\) and fixed coefficients falls into one of the classes, and all the classes are nonempty. We have
Theorem 2. There are no other algebraic movable singular points of equation (1.1) for \(k=2l+1\), apart from those distinguished in classes 2–6.
The proof is analogous to the proof of Theorem 1.
Remark 1. In classes 1 and 2 for \(k=2l\), and in classes 1–5 for \(k=2l+1\), besides the distinguished algebraic movable singular points there also exist such singular points that in their neighborhoods the asymptotic expansions \(y\) and \(z-z_0\) contain \(\ln x\).
Remark 2. It can be shown that for \(k=2l\) condition 3 and for \(k=2l+1\) condition 6 are necessary and sufficient in order that all movable singular points of the corresponding class of equations of the form (1.1) be algebraic.
References
- Golubev V. V. Lectures on the analytic theory of differential equations. GITTL, 1950.
- Erugin N. P. PMM, 16, issue 4, 465–486, 1952.
- Erugin N. P. Vestnik of Leningrad University, No. 7, issue 2, 60–70, 1956.
- Erugin N. P. Implicit functions. Publishing House of Leningrad University, 1956.
- Myachin V. F. Vestnik of Leningrad University, No. 7, issue 2, 1958.
Received by the editors
September 8, 1965
Vologda Pedagogical
Institute