AN INVESTIGATION OF A HOMOGENEOUS EQUATION OF THE FIRST ORDER NOT SOLVED WITH RESPECT TO THE DERIVATIVE

G. I. ZAPOROZHETS

The qualitative investigation of a homogeneous first-order equation solved with respect to the derivative is the subject of works by Shilov [1], Potlov [2], Lyatina [3], Sansone and Conti [4].

A homogeneous first-order equation of the second degree with linear coefficients was investigated by Picard [5], who, having solved it with respect to the derivative, then considered the resulting pair of two solved equations.

In the present investigation of the equation

\[ f\left(\frac{y}{x},\,\frac{dy}{dx}\right)=0 \tag{1} \]

a new method is applied, the essence of which is as follows: by introducing auxiliary variables, a plane curve is brought into correspondence with the differential equation; a connection is established between certain singularities of this curve, as well as singularities in its position relative to the coordinate system, and the properties of the integral curves; and thus the investigation of the differential equation over its whole domain of definition is reduced to the simpler problem of investigating a plane curve.

The expansions of the solutions of the equation under consideration obtained here in neighborhoods of various typical points, which serve as a basis for determining the properties of the integral curves, are analytic expressions for the solutions and as such may be used for solving other questions, in particular for the numerical integration of first-order equations not solved with respect to the derivative.

This method is also applicable to the investigation of the stability properties of the integral curves of the indicated equations, and, with certain restrictions, it extends also to a broader class of equations not solved with respect to the derivative.

§ 1. THE CURVE \(L\) CORRESPONDING TO EQUATION (1)

The properties of the solutions of equation (1) and their analytic expressions by means of series are closely connected with certain singularities of the function \(v(u)\), defined by the equation

\[ f(u,v)=0, \tag{2} \]

which is obtained from equation (1) as a result of the substitution

\[ \frac{y}{x}=u,\qquad \frac{dy}{dx}=v. \tag{3} \]

The graph of this function in a rectangular coordinate system will be called the curve \(L\) corresponding to equation (1).

Equation (2) will be assumed algebraic, i.e., the function \(f(u,v)\) is a polynomial in \(u\) and \(v\).

§ 2. ORDINARY POINTS OF THE FUNCTION \(v(u)\), THE CORRESPONDING EXPANSIONS OF SOLUTIONS OF EQUATION (1), AND THEIR PROPERTIES

If \(u=u_0\) is an ordinary point of the function \(v(u)\), then in some neighborhood of it this function is expanded in the convergent series

\[ v=v_0+a_1(u-u_0)+a_2(u-u_0)^2+\ldots+a_n(u-u_0)^n+\ldots, \tag{4} \]

and in the corresponding neighborhood of the line \(y=u_0x\), equation (1) can be written in the form

\[ \frac{d(y-u_0x)}{dx} = v_0-u_0+a_1\frac{y-u_0x}{x} +a_2\frac{(y-u_0x)^2}{x^2} +\ldots \tag{5} \]

Then, if the point \((x_0,y_0)\) lies on the line \(y=u_0x\), for the branch of the integral curve of equation (1) passing through it we obtain the expansion

\[ y=y_0+v_0(x-x_0)+\frac{a_1(v_0-u_0)}{2x_0}(x-x_0)^2+\ldots \tag{6} \]

or, when \(a_1=a_2=\ldots=a_{m-1}=0,\ a_m\ne0\), the expansion

\[ y=y_0+v_0(x-x_0)+ \frac{a_m(v_0-u_0)^m}{(m+1)x_0^m}(x-x_0)^{m+1} +\ldots \tag{6a} \]

From consideration of these expansions there follow the following propositions on the relation between the curve \(L\) and the integral curves of the corresponding equation*:

1) To a point \((u_0,v_0)\) of the curve \(L\) not lying on the bisector \(v=u\) there corresponds the line \(y=u_0x\), at whose points the corresponding branches of the integral curves have the same slope \(v_0\).

2) To a point \((u_0,v_0=u_0)\) of the curve \(L\) lying on the bisector \(v=u\) there corresponds the integral line \(y=u_0x\). (When \(v_0=u_0\) and \(y=u_0x\), equation (5) becomes an identity.)

3) If the curve \(L\) intersects the axis \(Ou\) at the point \((u_0\ne0;\,0)\), then at the points of the line \(y=u_0x\) the corresponding branches of the integral curves have an extremal value of the ordinate and a horizontal tangent. Moreover, if the curve \(L\) intersects the axis \(Ou\) while increasing (decreasing), then in a certain two-sided neighborhood of this line they are concave (convex) relative to the axis \(Ox\).

4) If at a point \((u_0,v_0\ne u_0)\) of the curve \(L\) the coordinate \(v\) has an extremal value, then the line \(y=u_0x\) is the geometric locus of the inflection points of the corresponding branches of the integral curves.

If the function \(v(u)\to v_0\) as \(u\to+\infty\), then for sufficiently large values of \(|u|\) it is expanded in a series of the form

\[ v=v_0+a_{-1}u^{-1}+a_{-2}u^{-2}+\ldots+a_{-n}u^{-n}+\ldots \tag{7} \]

* Here and below only the basic propositions necessary for the correct depiction of integral curves are indicated.

Accordingly, for the branch of the integral curve of equation (1) passing through the point \((0, y_0)\), in a neighborhood of the axis \(Oy\) we obtain the expansion

\[ y=y_0+v_0x+\frac{a_{-1}}{2y_0}\,x^2+\frac{a_{-2}-v_0a_{-1}}{3y_0^2}\,x^3+\ldots \tag{8} \]

or, if \(a_{-1}=a_{-2}=\ldots=a_{-(m-1)}=0,\ a_{-m}\ne0\), the expansion

\[ y=y_0+v_0x+\frac{a_{-m}}{(m+1)y_0^m}\,x^{m+1}+\ldots \tag{8a} \]

As a result of the investigation of these expansions, the details of which are omitted, we have:

5) If the curve \(L\) has a horizontal asymptote \(v=v_0\), then the integral curves at points of the axis \(Oy\) have the same angular coefficient \(v_0\). Moreover:

a) if, beginning with some sufficiently large value of \(|u|\), it approaches this asymptote while remaining on one side of it \([v\to v_0-0\) (or \(v\to v_0+0\)) both as \(u\to-\infty\) and as \(u\to+\infty]\), then the axis \(Oy\) is the locus of inflection of the corresponding branches of the integral curves;

b) if as \(u\to-\infty\) it approaches this asymptote from below (from above), and as \(u\to+\infty\)—from above (from below), then the corresponding branches of the integral curves in a neighborhood of the axis \(Oy\) are concave (convex) with respect to the origin; in this case, for \(v_0=0\), the axis \(Oy\) is the locus of points at which the branches of the integral curves have an extremal value of the ordinate.

§ 3. POSITION OF THE INTEGRAL CURVES OF EQUATION (1) IN A NEIGHBORHOOD OF ITS INTEGRAL STRAIGHT LINES

As was noted in § 2, to a point \((u_0, v_0=u_0)\) of the curve \(L\) there corresponds the integral straight line \(y=u_0x\) of equation (1).

The position of the integral curves of equation (1) in a neighborhood of its integral straight line that is not their envelope can be clarified (and classified) by investigating their convexity or concavity with respect to this line and to the origin.

To carry out such an investigation, write the series expansion for the branch of an integral curve of equation (1) passing through the point \((x_1,y_1)\), situated in a neighborhood of its integral straight line \(y=u_0x\),

\[ \begin{aligned} y={}&y_1+\bigl[u_0+a_1(u_1-u_0)+a_2(u_1-u_0)^2+\ldots\bigr](x-x_1)+{}\\ &+\frac{1}{2x_1}\bigl[a_1(a_1-1)(u_1-u_0)+a_2(3a_1-2)(u_1-u_0)^2+\ldots\bigr](x-x_1)^2+\ldots,\\ &\qquad u_1=\frac{y_1}{x_1}, \end{aligned} \tag{9} \]

and introduce the following conditions of convexity or concavity of curves:

A) The curve \(f(x,y)=0\) at its point \(M\) will be convex or concave with respect to the origin according as, at the point \(M\), the quantity

\[ A=\left(y-x\frac{dy}{dx}\right)\frac{d^2y}{dx^2} \tag{A} \]

has a positive or negative sign.

C) The curve \(f(x,y)=0\) at its point \(M\) will be convex or concave with respect to the straight line \(y=u_0x\), according as the quantity

\[ B=(y-u_0x)\left(1+u_0\frac{dy}{dx}\right)\frac{d^2y}{dx^2}. \tag{B} \]

has positive or negative sign at this point.

(The first condition may be obtained as the sign of the deviation of the center of curvature of the curve at the point \(M\) from the tangent to it at this point; the second—from the condition of convexity and concavity of the curve with respect to the axis \(Ox\) \((yy''\gtrless 0)\), by rotating the coordinate system through the angle \(\operatorname{arctg} u_0\).)

Using these conditions and the expansion (9), we obtain

\[ \operatorname{sign} A=-\operatorname{sign}\bigl[a_1+2a_2(u_1-u_0)+3a_3(u_1-u_0)^2+\ldots\bigr]; \]

\[ \operatorname{sign} B=\operatorname{sign}\bigl\{a_1(a_1-1)(1+u_0^2)+[a_2(3a_1-2)(1+u_0^2)+ \]

\[ {}+a_1^2u_0(a_1-1)](u_1-u_0)+\ldots\bigr\}. \]

From consideration of these equalities it follows:

If in some one-sided neighborhood of \(u_0\) one has: a) \(\dfrac{dv}{du}<0\), or b) \(0<\dfrac{dv}{du}<1\), or c) \(\dfrac{dv}{du}>1\), then in the corresponding one-sided neighborhood of the integral straight line \(y=u_0x\), inside the vertical angles between the straight lines \(y=(u_0-\varepsilon)x,\ y=u_0x\), or \(y=u_0x,\ y=(u_0+\varepsilon)x\), the corresponding branches of the integral curves in case a) are invariably convex with respect to the origin in the integral straight line, and, receding to infinity, they approach this straight line asymptotically; in case b) they are invariably concave with respect to the origin and this straight line; in case c) they are invariably convex with respect to the integral straight line and concave with respect to the origin, and therefore enter the origin, touching this straight line*.

In what follows, these three cases of the disposition of the integral curves in a neighborhood of the integral straight line will briefly be called: a) hyperbolic, b) parabolic, c) elliptic, and denoted briefly by: Г, П, Э.

If the integral curves are situated both in the left-sided and in the right-sided neighborhood of the integral straight line \(y=u_0x\), which occurs when \(L\) is situated both to the left and to the right of \(u_0\), then the following 7 cases are possible: Г—Г, П—П, Э—Э, Г—П, П—Г, П—Э, Э—П**. All these cases are shown in the appended drawings.

§ 4. POLES OF THE FUNCTION \(v(u)\) CORRESPONDING TO EXPANSIONS OF SOLUTIONS OF EQUATION (1) AND THEIR PROPERTIES

If \(u=u_0\) is a pole of order \(m\) of the function \(v(u)\), then in some neighborhood of it this function is expanded in a series of the form

\[ v=a_{-m}(u-u_0)^{-m}+a_{-(m-1)}(u-u_0)^{-(m-1)}+\ldots+a_{-1}(u-u_0)^{-1}+ \]

\[ {}+a_0+a_1(u-u_0)+a_2(u-u_0)^2+\ldots+a_n(u-u_0)^n+\ldots \tag{10} \]

* These cases were considered in investigation [1], but here the convexity or concavity of the integral curves was not taken into account, without which, in our opinion, their disposition with respect to the integral straight line cannot be described correctly.

** The cases Э—Г or Г—Э are possible when the integral straight line is the axis \(Oy\) (see Fig. 1).

In this case, for the branch of the integral curve of equation (1) passing through the point \((x_0, y_0)\) of the line \(y=u_0x\), we obtain the expansion

\[ x=x_0+\frac{1}{(m+1)a_{-m}x_0^m}(y-y_0)^{m+1} -\frac{a_{-(m-1)}}{(m+2)a_{-m}^2x_0^{m+1}}(y-y_0)^{m+2}+\ldots \tag{11} \]

or

\[ y=y_0+A_1(x-x_0)^{\frac{1}{m+1}} +A_2(x-x_0)^{\frac{2}{m+1}}+\ldots, \tag{11a} \]

whence it follows that \(x_0\) is an algebraic critical point. From consideration of these expansions we have:

6) If the curve \(L\) has a vertical asymptote \(u=u_0\), then the integral curves of equation (1) at the points of the line \(y=u_0x\) have a vertical tangent. Moreover:

a) if

\[ \lim_{u\to u_0-0}v=\lim_{u\to u_0+0}v=+\infty \quad(\text{or }-\infty), \]

i.e., if \(m\) is even, then the line \(y=u_0x\) is the locus of points of inflection of the corresponding branches of the integral curves;

b) if

\[ \lim_{u\to u_0-0}v=-\lim_{u\to u_0+0}v=+\infty \tag{\(\alpha\)} \]

or

\[ \lim_{u\to u_0-0}v=-\lim_{u\to u_0+0}v=-\infty, \tag{\(\beta\)} \]

i.e., if \(m\) is odd, then the line \(y=u_0x\) is a locus of points at which the corresponding branches of the integral curves have an extremal value of the abscissa; moreover, in a neighborhood of this line, in case \((\alpha)\) they are concave, and in case \((\beta)\) convex, with respect to the axis \(Oy\).

If the function \(v(u)\) has a pole of order \(m\) for \(u=\pm\infty\), then for sufficiently large values of \(|u|\) it expands in a series of the form

\[ v=b_mu^m+b_{m-1}u^{m-1}+\ldots+b_1u+b_0+b_{-1}u^{-1}+b_{-2}u^{-2}+\ldots, \tag{12} \]

and in the corresponding neighborhood of the axis \(Oy\), equation (1) may be replaced by the equation

\[ \frac{dx}{dy} = \frac{x^m}{ b_my^m+b_{m-1}xy^{m-1}+\ldots+b_0x^m+b_{-1}x^{m+1}y^{-1} +b_{-2}x^{m+2}y^{-2}+\ldots }. \tag{13} \]

Investigating these equalities, we conclude:

7) If the curve \(L\) has an infinitely distant point \((+\infty,+\infty)\) in the direction of the axis \(Ov\) (a pole of order \(m\), \(m=2,3,\ldots\)) or an oblique asymptote \(v=b_1u+b_0\) (a pole of order \(1\)), then, for the corresponding equation, the axis \(Oy\) is an integral straight line.

In this case, for the branch of the integral curve passing through the point \((x_1,y_1)\) in a neighborhood of the axis \(Oy\), we have the expansion

\[ \begin{aligned} x=x_1 &+\left[ \frac{1}{b_m}\left(\frac{x_1}{y_1}\right)^m -\frac{b_{m-1}}{b_m^2}\left(\frac{x_1}{y_1}\right)^{m+1} +\ldots \right](y-y_1) \\ &-\frac{1}{2y_1}\left[ \frac{m}{b_m}\left(\frac{x_1}{y_1}\right)^m -\frac{(m+1)b_{m-1}}{b_m^2}\left(\frac{x_1}{y_1}\right)^{m+1} +\ldots \right. \\ &\left. \qquad -\frac{m}{b_m^2}\left(\frac{x_1}{y_1}\right)^{2m-1} +\frac{(m+1)b_{m-1}}{b_m^3}\left(\frac{x_1}{y_1}\right)^{2m} -\ldots +\frac{mb_{m-1}}{b_m^3}\left(\frac{x_1}{y_1}\right)^{2m} -\ldots \right](y-y_1)^2+\ldots . \end{aligned} \tag{14} \]

and from conditions (A) and (B) (§ 3) we have:

\[ \operatorname{sign} A=-\operatorname{sign} b_m\left(\frac{y_1}{x_1}\right)^{m-1};\quad \operatorname{sign} B=-\operatorname{sign} x y' y''= \]

\[ = -\begin{cases} \operatorname{sign}\left[1-b_m\left(\dfrac{y_1}{x_1}\right)^{m-1}\right], & \text{for } m>1 \text{ and for } m=1,\ b_1\ne1,\\[1.2em] -\operatorname{sign} b_0\dfrac{y_1}{x_1}, & \text{for } m=b_1=1,\ b_0\ne0,\\[1.2em] -\operatorname{sign} b_{-(n+1)}\left(\dfrac{y_1}{x_1}\right)^n, & \text{for } m=b_1=1,\ b_0=b_{-1}=\ldots=b_{-n}=0,\\ & b_{-(n+1)}\ne0. \end{cases} \]

Fig. 1

Investigating the signs of the quantities \(A\) and \(B\), we conclude:

I. If, as \(u\to+\infty\) or as \(u\to-\infty\), the derivative \(\dfrac{dv}{du}\to +\infty(-\infty)\), then in the corresponding neighborhood of the axis \(Oy\) the integral curves have a parabolic (hyperbolic) arrangement.

II. If, as \(u\to+\infty\) or as \(u\to-\infty\), the derivative \(\dfrac{dv}{du}\to b_1\), then in the corresponding neighborhood of the axis \(Oy\), for \(b_1<0\) the integral curves are arranged hyperbolically, for \(0<b_1<1\) elliptically, and for \(b_1>1\) parabolically.

III. If, as \(u\to+\infty\) \((u\to-\infty)\), the derivative \(\dfrac{dv}{du}\to 1\), then in the corresponding neighborhood of the axis \(Oy\), for \(b_0>0\) the integral curves are arranged parabolically (elliptically), while for \(b_0<0\) they are arranged elliptically (parabolically).

IV. If \(b_0=0\) and, as \(u\to+\infty\) or as \(u\to-\infty\), the derivative

\[ \frac{dv}{du}\to 1-0(1+0), \]

then in the corresponding neighborhood of the axis \(Oy\) the integral curves have a parabolic (elliptic) arrangement.

In Fig. 1 the integral curves of the equation

\[ a^2x^2(y')^2-b^2y^2-a^2b^2x^2=0,\quad 0<b<a \]

and the corresponding curve \(L\) (a hyperbola) are shown.

The axis \(Oy\) is an integral straight line, in the left- and right-hand neighborhoods of which one system of branches has an elliptic arrangement, and the other a hyperbolic arrangement (item 7, case II); \(m_1\) and \(m_4\) are integral straight lines, in a two-sided neighborhood of which the integral curves have

parabolic disposition (§ 3); the axis \(Ox\) \((m_2, m_3)\)—geometric loci of inflection, in which one system of branches has slope \(v_2\), and the other—\(v_3\) (item 4).

Here, as in all the following drawings, the integral curves have no other singularities besides those indicated.

§ 5. ALGEBRAIC CRITICAL POINTS OF THE FUNCTION \(v(u)\), CORRESPONDING EXPANSIONS OF SOLUTIONS OF EQUATION (1) AND THEIR PROPERTIES

If \(u=u_0\) is an algebraic critical point of the function \(v(u)\), then in some neighborhood of it this function expands into a series of the form

\[ v=v_0+a_k(u-u_0)^{\frac{k}{m}}+a_{k+1}(u-u_0)^{\frac{k+1}{m}}+\ldots \tag{15} \]

For \(k<m\) the tangent to the curve \(L\) at the point \((u_0,v_0)\) is the line \(u=u_0\), for \(k=m\)—the line \(v-v_0=a_k(u-u_0)\), and for \(k>m\)—the line \(v=v_0\).

In the corresponding neighborhood of the line \(y=u_0x\), equation (1) can be represented in the form

\[ \frac{d(y-u_0x)}{dx} = v_0-u_0+ \frac{a_k (y-u_0x)^{\frac{k}{m}}}{x^{\frac{k}{m}}} + \frac{a_{k+1}(y-u_0x)^{\frac{k+1}{m}}}{x^{\frac{k+1}{m}}} +\ldots \tag{16} \]

Proceeding from this, for the branch of an integral curve passing through the point \((x_0,y_0)\) of the line \(y=u_0x\), we obtain the expansion

\[ y=y_0+v_0(x-x_0)+ \frac{m a_k (v_0-u_0)^{\frac{k}{m}}}{(m+k)x_0^{\frac{k}{m}}} (x-x_0)^{\frac{k+m}{m}} + \]

\[ + \frac{m a_{k+1}(v_0-u_0)^{\frac{k+1}{m}}}{(m+k+1)x_0^{\frac{k+1}{m}}} (x-x_0)^{\frac{k+m+1}{m}} +\ldots \tag{17} \]

Investigating the expansions of the functions \(v(u)\) and \(y(x)\), we arrive at the following conclusions:

8) To a point of return of the first kind \((u_0, v_0\ne u_0)\) of the curve \(L\), depending on the slope \(a\) of the tangent to it at this point, there correspond: a) for \(a=0\), points of return of the first kind; b) for \(a=\pm\infty\), points of inflection; and c) for \(a\ne0,\ a\ne\pm\infty\), points of return of the second kind of the integral curves, situated on the line \(y=u_0x\), the tangents at which are parallel to the line \(y=v_0x\).

9) To a point of return of the second kind \((u_0, v_0\ne u_0)\) of the curve \(L\) there correspond points of return of the second kind of the integral curves, situated on the line \(y=u_0x\), the tangents at which are parallel to the line \(y=v_0x\).

10) To a point \((u_0, v_0\ne u_0)\) of the curve \(L\), at which the coordinate \(u\) has an extremal value with tangent \(u=u_0\), there correspond points of return of the first kind of the integral curves, situated on the line \(y=u_0x\), the tangents at which are parallel to the line \(y=v_0x\).

11) To a point \((u_0, v_0\ne u_0)\) of the curve \(L\), at which the coordinate \(v\) has an extremal value with tangent \(v=v_0\) (for \(k>m\)), as in item 4, there correspond points of inflection of the integral curves, situated on the line \(y=u_0x\), the tangents at which are parallel to the line \(y=v_0x\).

Figure 2 shows the curve \(L\) (a hypocycloid)

\[ (u^2+v^2)(u^2+v^2+18a^2)+8au(3v^2-u^2)=27a^4, \]

corresponding to the equation

\[ x^4(y')^4+2x^2(y^2+12axy+9a^2x^2)(y')^2+y^4-8axy^3+18a^2x^2y^2-27a^4x^4=0, \]

and the integral curves of this equation. The straight lines \(m_1, m_2\) are the loci of returns of the second kind, whose tangents are parallel to the straight line \(y=v_1x\) (item 8, c), and the loci of returns of the second kind, whose tangents are parallel to the straight line \(y=v_2x\) (item 8, c); \(m_4\) and \(m_6\) are the loci of returns of the first kind, whose tangents are parallel to the axis \(Ox\) (item 10 and item 8, a); \(m_3\) and \(m_5\) are integral straight lines, in whose two-sided neighborhood the integral curves have, respectively, an elliptic and a hyperbolic arrangement (§ 3).

Fig. 2

If in the case under consideration \(v\to v_0\) as \(u\to\pm\infty\), then for sufficiently large values of \(|u|\) we have the expansion

\[ v=v_0+a_{-k}u^{-\frac{k}{m}}+a_{-(k+1)}u^{-\frac{k+1}{m}}+\cdots \tag{18} \]

Hence, for the branch of an integral curve of equation (1) passing through the point \((0,y_0)\), in a neighborhood of the axis \(Oy\) we obtain the expansion

\[ \begin{aligned} y&=y_0+v_0x+ \frac{ma_{-k}}{(k+m)y_0^{\frac{k}{m}}}\, x^{\frac{k+m}{m}} \\ &\quad+ \frac{ma_{-(k+1)}}{(k+m+1)y_0^{\frac{k+1}{m}}}\, x^{\frac{k+m+1}{m}}+\cdots . \end{aligned} \tag{19} \]

Studying these expansions, we conclude:

12) If the curve \(L\) has a horizontal asymptote \(v=v_0\), which, as \(u\to-\infty\) (or as \(u\to+\infty\)), it approaches both from below and from above, then the axis \(Oy\) is the locus of returns of the first kind of the integral curves, whose tangents are parallel to the straight line \(y=v_0x\). (As \(u\to-\infty\) they are located in a right-hand neighborhood, and as \(u\to+\infty\) in a left-hand neighborhood of the axis \(Oy\).)

Fig. 3

In Fig. 3 are shown the curve \(L\) (a conchoid)

\[ [(u-b)^2+v^2](v-a)^2-a^2v^2=0 \]

and the integral curves of the corresponding equation.

The axis \(Oy\) is the locus of return points of the first kind, whose tangents are parallel to the straight line \(y=v_5x\) (item 12); the straight lines \(m_2, m_3\) are the locus of inflection of the integral curves, at which one system of branches has angular coefficient \(v_2=0\) (item 8, b), and the other has \(v_3\) (item 11); \(m_1\) and \(m_4\) are integral straight lines, in a two-sided neighborhood of which the integral curves have a hyperbolic arrangement (§ 3).

For \(v_0=u_0\), equation (16) is satisfied by the function \(y=u_0x\). Therefore, if the point \((u_0,v_0)\) of the curve \(L\), corresponding to the algebraic critical point \(u_0\) of the function \(v(u)\), lies on the bisector \(v=u\), then, as in item 2, the corresponding equation (1) has the integral straight line \(y=u_0x\).

Moreover, for \(v_0=u_0\) and \(k<m\), for the branch of the integral curve passing through the point \((x_0,y_0)\) of the integral straight line \(y=u_0x\), from equation (16) we have the expansion

\[ y=y_0+u_0(x-x_0)+\left(\frac{m-k}{m}a_k\right)^{\frac{m}{m-k}}x_0^{\frac{k}{k-m}}(x-x_0)^{\frac{m}{m-k}}+\ldots \tag{20} \]

(For \(v_0=u_0\) and \(k>m\), equation (16) has only the unique solution \(y=u_0x\), which for \(x=x_0\) becomes \(y_0\).)

Analyzing this expansion, we arrive at the following conclusions:

13) If the curve \(L\) has a point \(Q(u_0,\ v_0=u_0)\) with tangent \(u=u_0\), then the integral straight line \(y=u_0x\) is the envelope of the corresponding branches of the integral curves.

At the same time, if \(Q\) is a) a return point of the first kind, or b) a return point of the second kind, or c) an inflection point, or d) a point with an extremal value of the coordinate \(u\), then the straight line \(y=u_0x\), for the branches of integral curves enveloped by it, will be: in case a) the locus of points of inflection, in case b) the locus of return points of the second kind, in case c) the locus of return points of the first kind, and in case d) an ordinary envelope touching the integral curves at their ordinary points.

The arrangement of the integral curves in a neighborhood of the integral straight line corresponding to the algebraic critical point \(u_0\) for \(k>m\) depends on the form of the curve \(L\) near the point \((u_0, v_0=u_0)\) and is easily determined from the criteria indicated in § 3. Omitting a detailed description of the various possible cases here, we shall explain them by examples.

In Fig. 4 the curve \(L\) (an astroid)

\[ (u+v)^{2/3}+(u-v)^{2/3}=a^{2/3} \]

Fig. 4

corresponds to the equation

\[ 8x^6(y')^6+3x^4(5a^2x^2+8y^2)(y')^4+6x^2(a^4x^4-13a^2x^2y^2+4y^4)(y')^2- \]

\[ -a^6x^6+6a^4x^4y^2+15a^2x^2y^4+8y^6=0. \]

The lines \(m_1, m_2\) are geometric loci of points of return of the second kind, the tangents to which are parallel to the line \(y=v_2x\) (item 8, c), and an integral line, in a right-hand neighborhood of which one system of branches is arranged elliptically, and the other parabolically (§ 3); the lines \(m_3\) and \(m_6\) are geometric loci of points of return of the first kind, the tangents to which are parallel to the axis \(Ox\) (item 10); the axis \(Ox\) is a geometric locus of points of inflection, in which one system of branches has angular coefficient \(v_4\), and the other \(v_5\) (item 11); the lines \(m_7, m_8\) are geometric loci of points of return of the second kind, the tangents to which are parallel to the line \(y=v_7x\) (item 8, c), and an integral line, in a left-hand neighborhood of which one system of branches is arranged elliptically, and the other parabolically (§ 3).

Fig. 5

In Fig. 5 are given the curve \(L\)

\[ u^4+u^2v^2-2u^2v-uv^2+v^2=0 \]

and the integral curves of the corresponding equation

\[ (x^2y^2-x^3y+x^4)(y')^2-2x^2y^2y'+y^4=0. \]

The axis \(Ox\) is an integral line, in a right-hand neighborhood of which the two systems of branches of the integral curves have a parabolic arrangement (§ 3); the line \(m_4\) is an ordinary envelope (item 13, d); \(m_2\) is an integral line, in a two-sided neighborhood of which the corresponding branches have an elliptic arrangement (§ 3); the line \(m_3\) is a geometric locus of points of inflection (item 11).

§ 6. CRITICAL POLES OF THE FUNCTION \(v(u)\) CORRESPONDING TO EXPANSIONS OF SOLUTIONS OF EQUATION (1) AND THEIR PROPERTIES

If the function \(v(u)\) has a critical pole \(u_0\), then in some neighborhood of it it expands into a series of the form

\[ v=a_{-k}(u-u_0)^{-\frac{k}{m}}+a_{-(k-1)}(u-u_0)^{-\frac{k-1}{m}}+\ldots+a_0+ \]

\[ +a_1(u-u_0)^{\frac{1}{m}}+a_2(u-u_0)^{\frac{2}{m}}+\ldots \tag{21} \]

Hence, for a branch of the integral curve of equation (1) in the corresponding neighborhood of the point \((x_0,y_0)\) of the line \(y=u_0x\), we have the expansion

\[ y=y_0+u_0(x-x_0)+\left(\frac{k+m}{m}a_{-k}\right)^{\frac{m}{k+m}}x_0^{\frac{k}{k+m}}(x-x_0)^{\frac{m}{k+m}}+\ldots \tag{22} \]

Investigating these expansions, we conclude:

14) If the curve \(L\) has an asymptote \(u=u_0\), and if \(u\to u_0-0\) (or \(u\to u_0+0\)) both as \(v\to-\infty\) and as \(v\to+\infty\), then the line \(y=u_0x\) is a g.m.t. of return of the first kind of the integral curves whose tangents are parallel to the axis \(Oy\).

To the critical pole \(u=\pm\infty\) of the function \(v(u)\) there corresponds the expansion

\[ v=a_k u^{\frac{k}{m}}+a_{k-1}u^{\frac{k-1}{m}}+\cdots+a_0+a_{-1}u^{-\frac{1}{m}}+a_{-2}u^{-\frac{2}{m}}+\cdots, \tag{23} \]

according to which the curve \(L\) has an infinitely distant point \((\pm\infty,\pm\infty)\) in the direction of the line \(u=u_0\), or \(v=a_k u\), or \(v=v_0\), depending on whether \(k>m\), or \(k=m\), or \(k<m\).

In a neighborhood of the axis \(Oy\), equation (1), corresponding to the expansion (23), has the form

\[ \frac{dx}{dy} = \frac{1}{a_k}\left(\frac{x}{y}\right)^{\frac{k}{m}} - \frac{a_{k-1}}{a_k^2} \left(\frac{x}{y}\right)^{\frac{k+1}{m}} + \frac{a_{k-1}^{2}-a_k a_{k-2}}{a_k^3} \left(\frac{x}{y}\right)^{\frac{k+2}{m}} +\cdots \tag{24} \]

Therefore:

15) If the curve \(L\) has an infinitely distant point \((\pm\infty,\pm\infty)\), then for the corresponding equation the axis \(Oy\) is an integral line.

Fig. 6

From equation (24), for \(k<m\), for the branch of the integral curve passing through the point \((0,y_0)\) of the axis \(Oy\), we have the expansion

\[ y = y_0 + \frac{m a_k}{m-k}\, y_0^{\frac{k}{m}} x^{\frac{m-k}{m}} + \frac{m a_{k-1}}{m-k+1}\, y_0^{\frac{k-1}{m}} x^{\frac{m-k+1}{m}} +\cdots \tag{25} \]

(For \(k\ge m\), equation (24) has the unique integral line \(x=0\), passing through the point \((0,y_0)\) of the axis \(Oy\).)

From this it follows:

16) If the curve \(L\) has an infinitely distant point \((\pm\infty,\pm\infty)\) in the direction of the line \(v=v_0\), then the axis \(Oy\) is the envelope of the corresponding branches of the integral curves.

Moreover, if a) \(v\to-\infty\) (or \(v\to+\infty\)) both as \(u\to-\infty\) and as \(u\to+\infty\), or b) \(v\to-\infty\) (or \(v\to+\infty\)) as \(u\to-\infty\), and \(v\to+\infty\) (or \(v\to-\infty\)) as \(u\to+\infty\), or c) \(u\to-\infty\) (or \(u\to+\infty\)) both as \(v\to-\infty\) and as \(v\to+\infty\), then the axis \(Oy\), for the branches of the integral curves enveloped by it, will be: in case a), a g.m.t. of inflection; in case b), a g.

i.e. of a return point of the first kind, and in case c) a g.m.t. with an extremal (zero) value of the abscissa.

In Fig. 6 the curve \(L\) (a cubical parabola) \(v^3=u\) corresponds to the equation \(x(y')^3=y\).

The axes \(Ox\) and \(Oy\) are envelopes—g.m.t.’s of return of the first kind of the integral curves enveloped by them (item 16, b and item 13, c); \(m_1\) and \(m_3\) are integral straight lines, in whose two-sided neighborhoods the integral curves have a parabolic disposition (§ 3).

Fig. 7

In Fig. 7 the integral curves of the equation

\[ x^3(y')^6-3x^2y(y')^4-2x^2y(y')^3+3xy^2(y')^2-6xy^2y'+y^2(x-y)=0 \]

and the corresponding curve \(L\)

\[ (v^3+3uv-u)^2-u(3v^2+u)^2=0 \]

are shown.

The axis \(Oy\) is the ordinary envelope of the corresponding branches (item 16, c); the axis \(Ox\) is an envelope—a g.m.t. of return of the second kind of the branches it envelops (item 13, b); the straight line \(m_2\) is a g.m.t. of inflection (item 11); the straight line \(m_4\) is a g.m.t. with an extremal value of the ordinate and a horizontal tangent (item 3); \(m_3\) and \(m_5\) are integral straight lines, in whose two-sided neighborhoods the integral curves have respectively a hyperbolic and a parabolic disposition (§ 3).

Investigating the disposition of the integral curves in a neighborhood of the integral straight line \(x=0\), corresponding to the point at infinity \((\pm\infty,\pm\infty)\) of the curve \(L\) for \(k>m\), according to the signs of the quantities \(A\) and \(B\) (§ 3),

\[ \operatorname{sign} A=-\operatorname{sign}\frac{dv}{du},\qquad \operatorname{sign} B=\operatorname{sign} v(u-v)\frac{dv}{du}, \]

we conclude:

V) If \(u\to-\infty\) (or \(u\to+\infty\)) both as \(v\to-\infty\) and as \(v\to+\infty\), then in the corresponding one-sided neighborhood of the axis \(Oy\) one system of branches of the integral curves is disposed hyperbolically, and the other parabolically. The other cases for \(k>m\) are analogous to cases I, § 4.

VI) For \(k=m\) and \(a_k\ne1\), the disposition of the integral curves in a neighborhood of the axis \(Oy\) is similar to that described in case II, while for \(a_k=1\) it is similar to that described in cases III and IV, § 4. But if, as \(u\to-\infty\) (or as \(u\to+\infty\)), two branches of the curve \(L\) recede to infinity, then in the corresponding one-sided neighborhood of the axis \(Oy\) there are two systems of branches of the integral curves. This also includes the case when the curve \(L\) has an oblique asymptote, to which, as \(u\to-\infty\) (or as \(u\to+\infty\)), one of its branches approaches from below and the other from above.

In Fig. 8 the curve \(L\)

\[ [u^4-3u^3(v-au)-(v-au)^3]^2=u^3[u^3+3(v-au)^2]^2,\qquad a>1 \]

and integral curves of the equation (of the 6th degree) to which it corresponds.

The axis \(Oy\) is an integral straight line, in whose left-hand neighborhood one system of branches is arranged hyperbolically and the other parabolically (item 15, case V); the axis \(Ox\) is an integral straight line, in whose right-hand neighborhood two systems of branches have an elliptic arrangement (§ 3); the straight line \(m_2\) is, so to speak, an inflection point (item 11); \(m_3\) is an integral straight line, in whose two-sided neighborhood the corresponding branches have a hyperbolic arrangement (§ 3); the straight line \(m_4\) is, so to speak, one at which the corresponding branches have an extremal value of the ordinate and a horizontal tangent (item 3).

Fig. 8

§ 7. Points of contact, nodal and isolated points of the curve \(L\)

If the curve \(L\) has a nodal point \((u_0, v_0 \ne u_0)\), at which \(n\) of its branches intersect, then in a neighborhood of this point each of them corresponds to an expansion of the form (4) or of the form (15).

According to the conclusions of §§ 2 and 5, the systems of branches of the integral curves in a neighborhood of each point \((x_0, y_0)\) of the straight line \(y = u_0x\) corresponding to these intersecting branches of the curve \(L\) are determined by expansions of the form (6) or of the form (17), in which the first terms coincide:

\[ y = y_0 + v_0(x - x_0). \]

Therefore, through each point of the straight line \(y = u_0x\) there pass \(n\) distinct integral curves of the corresponding equation, which have a common tangent.

To a point of contact \((u_0, v_0 \ne u_0)\) of the curve \(L\), at which its branches touch one another, there correspond expansions of the same form, but with a larger number of common initial terms. Therefore, at points of the straight line \(y = u_0x\) the corresponding branches of the integral curves have contact whose order is one greater than the order of contact of the branches of the curve \(L\).

For \(v_0 = u_0\), the corresponding equation has the integral straight line \(y = u_0x\), in whose neighborhood the branches of the integral curves have one or another arrangement depending on the form of the branches of the curve \(L\) in a neighborhood of its indicated special points.

In general, in a neighborhood of the straight line \(y = u_0x\), corresponding to a point of contact or a nodal point of the curve \(L\), the integral curves of the corresponding equation possess the aggregate of those properties which, separately, correspond to it as to a point lying on each of the tangent or intersecting branches of the curve \(L\).

Thus, in Fig. 9 the curve \(L\)

\[ (u^2 + v^2)^2 - av(3u^2 - v^2) = 0, \]

which has a triple nodal point, corresponds to the equation

\[ x^4(y')^4 + ax^4(y')^3 + 2x^2y^2(y')^2 - 3ax^2y^2y' + y^4 = 0. \]

The axis \(Ox\) is an integral straight line \((m_5)\), in a left-hand neighborhood of which two systems of branches are situated hyperbolically and one—

Fig. 9

elliptically, while in the right-hand one system of branches is situated hyperbolically, another elliptically, and a third parabolically (§ 3, cases a, b, c), and, in addition, it is a g.m.p. of inflection \((m_4,\) item 11); the straight lines \(m_1, m_3, m_6\) and \(m_9\) are g.m.p.’s of return of the first kind (item 10);

Fig. 10

the straight lines \(m_2\) and \(m_8\) are g.m.p.’s of inflection (item 11); \(m_7\) is an integral straight line, in a two-sided neighborhood of which the corresponding branches have a parabolic arrangement (§ 3).

In Fig. 10 the curve \(L\) is shown

\[ v^2 + (u - 1)(u - 3)(u - 2)^4 = 0, \]

having a point of tangency of the 1st order \((2; 0)\), and the integral curves of the corresponding equation

\[ x^5(y')^2+(y-x)(y-3x)(y-2x)^4=0. \]

On the line \(m_4\) there are located points of inflection of two systems of branches, at which they have tangency of the 2nd order and a horizontal tangent (item 11); the lines \(m_1\) and \(m_7\) are, in the terminology of the theory, return points of the first kind with a horizontal tangent (item 10); the lines \(m_2, m_3\) and \(m_5, m_6\) are, in the terminology of the theory, points of inflection (item 11).

To an isolated point \((u_0, v_0=u_0)\) of the curve \(L\) there corresponds a special integral straight line \(y=u_0x\), which cannot be an envelope, and in its neighborhood the integral curves can have neither hyperbolic, nor parabolic, nor elliptic disposition; while for \(v_0\ne u_0\) no properties of the integral curves correspond to it.

For example, the equation

\[ x^4(y')^2-2bx^4y'+(2x-y)^2(8x^2-y^2)+b^2x^4=0, \]

to which there corresponds the curve \(L\)

\[ (v-b)^2=(u-2)^2(u^2-8), \]

having the isolated point \((2,b)\), for \(b=2\) has the special integral straight line \(y=2x\). For \(b\ne2\) the integral curves of this equation will be the same as for the simpler equation, to which there corresponds the curve \(L_1\)

\[ (v-b)^2=u^2-8 \]

which differs from \(L\) by the absence of an isolated point.

References

1. G. E. Shilov, Uspekhi Mat. Nauk, 5, no. 5, 1950, pp. 193–203.
2. V. V. Potlov, Scientific Notes of the Ryazan State Pedagogical Institute, 15, 1957, pp. 67–76.
3. V. V. Potlov, Scientific Notes of the Ryazan State Pedagogical Institute, 24, 1960, pp. 119–125.
4. V. V. Potlov, Scientific Notes of the Ryazan State Pedagogical Institute, 24, 1960, pp. 127–137.
5. L. S. Lyagina, Uspekhi Mat. Nauk, 6, no. 2, 1951, pp. 171–183.
6. G. Sansone, R. Conti, Scritti mat. onore Filippo Sibirani, Bologna, 1957, pp. 243–260.
7. E. Picard, Traité d’Analyse, vol. III, 1896, G—V. Paris.
8. A. V. Pkhakadze and A. A. Shestakov, Mat. Sb., 49 (91): 1, 1959, pp. 3–12.

Received by the editors
November 13, 1964

All-Union Correspondence Institute
of Railway Transport Engineers