Adv Syst Sci Appl 2022; 04:51–64
Published online at https://ijassa.ipu.ru.

Oscillating and Proper Solutions of Singular Quasi-Linear
Differential Equations

Natalya G. Pavlova1,2,3 Alexey O. Remizov2*

1V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
2Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia

3RUDN University, Moscow, Russia

Abstract: We study the behaviour of solutions of quasi-linear differential equations of the
second order at their singular points, where the coefficient of the second-order derivative vanishes.
We consider solutions entering a singular point either with definite tangential direction (proper
solutions) or without definite tangential direction (oscillating solutions). Equations of this type
appear in many problems arising in analysis, geometry, dynamical systems theory, and physics.
First, we prove that a generic equation of the considered type has no oscillating solutions. Then
we concentrate on proper solutions, which can enter a singular point in admissible tangential
directions only. Great attention is paid to second-order differential equations, whose right-hand
sides are cubic polynomials by the first-order derivative. We obtain local representations for
solutions of such equations in a form similar to Newton–Puiseux series – series with fractional
exponents (and, in a special case, with logarithmic terms).

Keywords: differential equations, singular points, normal forms, oscillating solutions

1. INTRODUCTION

1.1. A brief look at quasi-linear equations
The study of singular quasi-linear differential equations goes back to the middle of the XIX
century. One of the first examples is the equation

xmy′ = f(x, y) (1.1)

named after Briot and Bouquet (French mathematicians of the 19th century), who carried
out a detailed study of such equations for analytic functions f ; the main achievement of this
study was representation of solutions in the form of power series with fractional exponents.
In that time, the theory developed by Briot and Bouquet seemed so significant that one of the
problems at the famous prize competition (1885) sponsored by Oscar II†, was to develop a
comparable theory for non-linear equations of the general form F (x, y, y′) = 0. In the same
year, Poincaré published his third memoir “On Curves Defined by Differential Equations”,
where he gave a start to the geometric approach in the theory of differential equations.‡

∗Corresponding author: alexey-remizov@yandex.ru
†King of Sweden (1872–1907) and Norway (1872–1905). In his youth he studied at Uppsala University, where he
distinguished himself in mathematics.
‡It is worth observing that Poincaré was awarded Oscar II prize for solution of another problem.



52 N. G. PAVLOVA, A. O. REMIZOV

Later on, the interest to singular quasi-linear differential equations was motivated either
by pure theoretical aspects or by various applications. For example, equations

∆(x, y)y′′ = M(x, y, y′) (1.2)

appear in many problems arising in analysis, geometry, dynamical systems theory, and
physics. Here one can name the Bessel equation and the Gaussian hypergeometric equation, to
name a few. We also notice equations appearing in two recent works: a generalized Ginzburg-
Landau model for liquid crystals [8] and Newton’s aerodynamic problem [16].
Definition 1.1:
A point q0 = (x0, y0) is called a regular point of equation (1.2) if ∆(x0, y0) ̸= 0, and it is
called a singular point of equation (1.2) if ∆(x0, y0) = 0.

Among equations (1.2), a great attention is paid to the case that the right-hand side is a
cubic polynomial in y′, that is,

M(x, y, y′) =
3∑

i=0

µi(x, y)(y
′)i. (1.3)

An attention to equations (1.2), (1.3) is motivated by their role in physics and geometry,
for instance, the description of various geometric structures (geodesic flows in affine or
projective connection, etc.). Equations of this class were studies by Sophus Lie, A. Tresse,
J. Liouville, E. Cartan, etc.

Among others, an important case is the Levi–Civita connection, which is generated by the
metric tensor

ds2 = a(x, y) dx2 + 2b(x, y) dxdy + c(x, y) dy2. (1.4)
Geodesics in the Levi–Civita connection are solutions of equation (1.2), (1.3), where

∆ = ac− b2

and the coefficients of the cubic polynomial M are polynomials on the coefficients a, b, c and
their first-order derivatives:

µ0 = a(ay − 2bx) + axb,

µ1 = b(3ay − 2bx) + axc− 2acx,

µ2 = b(2by − 3cx) + 2ayc− acy,

µ3 = c(2by − cx)− bcy.

(1.5)

Here ∆ vanishes at points where the quadratic form (1.4) degenerates. In Riemannian
geometry, such is not the case, since the inequality ∆ > 0 holds true everywhere. However,
metrics (quadratic forms) with varying signature generically appear on surfaces embedded
into pseudo-Euclidean spaces: the metric induced on a surface into pseudo-Euclidean space
degenerates at those points where the surface tangents the light cone of the ambient space.
Singularities of the geodesic equation appearing at degenerate points of the metric (1.4) are
studied in the series of papers [22] – [19].

1.2. Phenomenon of oscillating solutions
A. F. Filippov showed [5] that a system of ordinary differential equations

F (x, y, p) = 0, p = dy/dx, (1.6)

where x ∈ R1, y = (y1, . . . , yn) ∈ Rn, p = (p1, . . . , pn), and F : R2n+1 → Rn is a smooth
(by smooth we mean C∞, if the otherwise is not stated) vector function, may have

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OSCILLATING AND PROPER SOLUTIONS 53

solutions without definite tangential direction at a certain point. This motivates the following
terminology.

A solution y(x) of system (1.6) is called oscillating at a point (x0, y0), if y(x) is a vector
function differentiable on the interval (x0, x0 + δ) or (x0 − δ, x0), δ > 0, such that y(x) → y0
but y′(x) has neither finite not infinite limit as x → x0. A solution y(x) is called proper at a
point (x0, y0), if y′(x) has limit (finite or infinite) as x → x0.

Let J1 be the 1-jet space of smooth functions y(x), with the coordinates (x, y, p). Filippov
showed that if for a point T0 ∈ J1 there exists T ′

0 ∈ J1 such that x0 = x′
0, y0 = y′0, p0 ̸= p′0,

and the matrix Fp degenerates at T ′
0, then besides a unique proper solution passing through

(x0, y0) with the tangential direction p0, system (1.6) may have oscillating solutions, which
pass through (x0, y0) and even have definite tangential direction p0 at (x0, y0).

The following example is taken from [5]. Consider solutions of the system{
p1(1− p21 − p22) + 8xy1 + 4y2 = 0,

p2(1− p21 − p22) + 8xy2 − 4y1 = 0,
(1.7)

that pass through the origin with the tangential direction pi = 0, that is, satisfy the conditions
yi(0) = y′i(0) = 0, i = 1, 2. The matrix Fp is non-degenerate at the point T0 = 0, and
system (1.7) has a unique proper solution yi(x) ≡ 0, which obviously satisfies the required
conditions. However, system (1.7) has an infinite number of oscillating solutions given by the
formula

y1(x) = x2 cos(x−1 + c), y2(x) = x2 sin(x−1 + c), c = const,

for x ̸= 0, and yi(0) = 0. The both derivatives y′i(x) are zero at x = 0, but the limits y′i(x) as
x → 0 do not exist.

On the other hand, Filippov showed that systems with oscillating solutions are a sort of
exception. Given a point q0 = (x0, y0) ∈ Rn+1, we define the set

Q(q0) = {T = (q0, p) ∈ J1 : F (T ) = 0, det(Fp(T )) = 0}.
Theorem 1.1 ( [5]):
Assume that at least one of the following conditions holds true:

1. p0 /∈ coQ(q0), where co denotes the convex hull,
2. the set Q(q0) is at most countable (i.e., countable, or finite, or empty).
Then system (1.6) has no solution oscillating at q0 with tangential direction p0 at q0.

Generically, the condition 2 holds true at all points, and system (1.6) has no oscillating
solutions.

Although Theorem 1.1 has a very general character, it gives no information for some
special classes of systems. For instance, it gives only a trivial result for quasi-linear implicit
systems

A(x, y) p = b(x, y), p = dy/dx, (1.8)
where A is a n× n matrix depending on (x, y), b is a vector function. It is worth observing
that system (1.8) are of great interest due to various applications, especially in electrical
circuit theory. See, e.g., the papers [27] – [21] and the references therein).

Theorem 1.1 guarantees the absence of oscillating solution at all points where detA ̸= 0,
which obviously follows from the fact that then system (1.8) is locally equivalent to the
explicit system p = A−1(x, y)b(x, y). However, it says nothing about points where the
matrix A degenerates, since Q(q0) = {q0} × Rn if detA(q0) = 0, and the both conditions
in Theorem 1.1 fail.

1.3. The aim, scope and structure of the paper
The main emphasis of this paper is on oscillating solutions of quasi-linear second-order
differential equations (1.2). It should be remarked that in all papers mentioned above (as well

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54 N. G. PAVLOVA, A. O. REMIZOV

as in other works known to us) only proper geodesics were considered, while the possibility
of oscillating geodesics is not studied yet. In the present paper, we shall fill this gap.

The paper is organized as follows. In Section 2, we establish a necessary condition for
quasi-linear equation (1.2) to have an oscillating solution – Theorem 2.1, which plays the
same role as that played by Theorem 1.1 for non-linear systems (1.6). This condition shows
that a generic equation (1.2) with M analytic in p has no oscillating solutions, but proper
solutions only. In particular, equation (1.2) with a cubic polynomial (1.3) that describes
geodesics in the Levi–Civita connection generated by the metric tensor (1.4) has no oscillating
solutions if ∆, d∆ do not vanish simultaneously.

In Section 3, we briefly describe proper solutions of equation (1.2) with the right-hand
side (1.3) entering its generic singular points. Proper solutions cannot enter singular points in
arbitrary tangential directions, but only at so-called admissible directions p that correspond
to real roots of the cubic polynomial M in p. To every admissible direction p, where p is
a primer root of M , corresponds either a unique smooth solution or an infinite family of
solutions with the common tangential direction and power singularity. Theorem 3.3 gives the
local representation of such solutions in a form similar to Newton–Puiseux series. This can
be considered as a far development of the results obtained by Poincaré and his predecessors
(Briot, Bouquet, Fuchs, etc.) for differential equations (1.1). However, in studying second-
order equations we encounter a special difficulty that does not appear in the case of the first
order equations: vector fields with non-isolated singular points. A systematic study of such
vector fields begins in the end of XX century, the first known works are [25, 28]. A brief
survey of results on the local classification of such fields can be found in [24] (Appendix).

In Section 4, we present a natural generalization of the results obtained before for quasi-
linear differential equations of higher orders. Finally, we should remark the closely related
papers [9] – [29] and especially [26].

2. OSCILLATING SOLUTIONS

In this section, we shall deal with differential equation (1.2). Following the tradition to denote
derivatives by single letters going back to Gaspard Monge, we write this equation in the
following form:

∆(x, y)
dp

dx
= M(x, y, p), p = dy/dx. (2.9)

The functions ∆(x, y), M(x, y, p) are supposed to be smooth.
Denote the locus of singular points of equation (2.9) by Γ. Generically, the set

Γ = {(x, y) : ∆(x, y) = 0}

is a curve on the (x, y)-plane, which is called singular curve or degenerate curve of the
equation. We shall assume this in what follows.

Consider the initial value problem for equation (2.9) with the initial condition y(x0) = y0.
If the point q0 = (x0, y0) is regular, for every p0 this problem has a unique solution satisfying
the additional condition p(x0) = p0, which is defined and smooth on a real interval including
the point x0, and it has no solutions without lim p(x) as x → x0. The situation becomes
different if q0 is singular (see the examples below). This makes sense to refine the notion
of solutions for the case of singular points. Throughout this paper, we use the following
notation: if I is an open real interval, then I denotes the closed interval (segment) with the
same endpoints.

Definition 2.1:
Let q0 = (x0, y0) ∈ Γ. A solution y(x) of equation (2.9) enters the point q0 if the function
y(x) satisfies the following conditions:

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OSCILLATING AND PROPER SOLUTIONS 55

1. It is continuous on a non-empty segment Iε with endpoints x0 and x0 + ε, where ε can
be either positive or negative, y(x0) = y0.

2. The function y(x) is differentiable and it satisfies (2.9) on the open interval Iε.
3. The graph of y(x) has no common points with Γ except for q0.
If in addition to the above conditions, the derivative p(x) has a (finite or infinite) limit as

x → x0, the solution y(x) is called proper. Otherwise the solution y(x) is called oscillating
at the point q0. See Fig. 2.1.

Definition 2.2:
A solution y(x) is called passing through the point q0 if the function y(x) is differentiable
and it satisfies (2.9) at all points of an open interval I that contains x0, y(x0) = y0, and the
graph of y(x), x ∈ I , intersects Γ at the point q0 only.

The significance of the given definitions will become clear in the examples presented
below. Obviously, every solution passing through q0 is the union of two solutions entering q0.
It is worth observing that solutions from Definitions 2.1, 2.2 are also called one-sided or two-
sided solutions, respectively (for example, in [26]). In this paper, we prefer the terminology
given above, which shows the geometric flavour of these notions.

Example 2.1:
The parabolas y = αx2, α = const, are integral curves of the second-order equation

2y
dp

dx
= p2, p = dy/dx,

passing through the origin. However, the graphs of functions defined by the formula

f(x) =

{
α1x

2, x ≤ 0,

α2x
2, x > 0,

are also integral curves, and all such functions are solution of the given equation passing
through the origin. To avoid such ambiguity, it is sufficient to use the notion of a solution
entering a point defined above. Then solutions entering the origin are the branches of the
parabolas y = αx2, α = const, and only them.

In the previous example, all solutions are proper. Now we give two examples of equations
(2.9) that possess oscillating solutions.

Example 2.2:
The equation

x4 dp

dx
= 2x3p− (2x2 + 1)y, p = dy/dx,

has the family of solutions defined by the formula

y = x2(α cosx−1 + β sinx−1) for x ̸= 0, α, β = const, (2.10)

and zero for x = 0. These functions are differentiable on the whole real axis, but their
derivatives are not continuous at the origin except for α = β = 0. According to the given
definitions, formula (2.10) gives a family of solutions passing through the origin with the
tangential direction p = 0. For α = β = 0, we have the regular solution y = 0, while for all
others α, β the solutions (2.10) are oscillating at the origin.

Example 2.3:
The equation

x2 dp

dx
= xp− 2y, p = dy/dx,

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56 N. G. PAVLOVA, A. O. REMIZOV

has the family of solutions

y = x(α cos ln |x|+ β sin ln |x|), α, β = const, (2.11)

entering the origin. Except for α = β = 0, all these solutions are oscillating at the origin and
(unlike the previous example) they have no definite tangential directions at the origin.

Moreover, using formula (2.11), one can construct an equation of the form (2.9) with
oscillating solution, whose right-hand side is a polynomial of arbitrary degree in p. For
instance, one can see that the function (2.11) with α = β = 1 is a solution of the first-order
equations fi(x, y, p) = 0, where

f2 = (xp)2 − 2xyp+ 2(y2 − x2),

f3 = (xp)3 + y(xp)2 − 2xp(x2 + 2y2) + 6y(y2 − x2).

Therefore, y = x(cos ln |x|+ sin ln |x|) is an oscillating solution of the second-order
equations

x2 dp

dx
= xp− 2y + fi(x, y, p), p = dy/dx.

We presented several examples demonstrating the existence of oscillating solutions. The
following theorem shows that oscillating solutions do not appear generically.

Theorem 2.1:
Let q0 ∈ Γ and M(q0, p) be an analytic function not identically zero. Then equation (2.9) has
no oscillating solutions entering q0.

Proof
Assume that the above conditions hold true, but equation (2.9) has an oscillating solution
y(x) that enters q0 without definite tangential direction. By the definition, y(x) is defined
on a segment Iε = [x0, x0 + ε] and ∆(x, y(x)) ̸= 0 at all its inner points: x0 < x < x0 + ε.
Without loss of generality, assume that ∆(x, y(x)) > 0 for all x such that x0 < x < x0 + ε
(otherwise we multiply the both sides of (2.9) by −1). From the absence of the limit y′(x) as
x → x0 it follows that there exist two sequences

x′
n → x0 + 0 and x′′

n → x0 + 0

such that
p(x′

n) → p′ and p(x′′
n) → p′′, p′ ̸= p′′.

For definiteness, assume that p′ < p′′. Since the function y′(x) is continuous on the
interval x0 < x < x0 + ε, for every p∗ ∈ (p′, p′′) there exist two sequences

ξn → x0 + 0 and ξ′n → x0 + 0

such that

lim
n→∞

p(ξn) = lim
n→∞

p(ξ′n) = p∗,
dp

dx
(ξn) > 0,

dp

dx
(ξ′n) < 0. (2.12)

See Fig. 2.1 (right). Substituting the solution y(x) into (2.9), at the points x = ξn and x = ξ′n
we have the equalities

∆(ξn, y(ξn))
dp

dx
(ξn) = M(ξn, y(ξn), p(ξn)), (2.13)

∆(ξ′n, y(ξ
′
n))

dp

dx
(ξ′n) = M(ξ′n, y(ξ

′
n), p(ξ

′
n)). (2.14)

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OSCILLATING AND PROPER SOLUTIONS 57

The right-hand sides of both equalities (2.13), (2.14) have the finite limit M(x0, y0, p∗),
whence their left-hand sides have the same limit M(x0, y0, p∗). On the other hand,
∆(ξn, y(ξn)) and ∆(ξ′n, y(ξ

′
n)) are positive for every n, and from (2.12) it follows that the

left-hand sides of equalities (2.13), (2.14) have different signs. Therefore, their limit

M(x0, y0, p∗) = 0.

Thus, we proved that M(q0, p∗) = 0 for every p∗ ∈ (p′, p′′). Since the function M(q0, p)
is analytic in p, this implies M(q0, p) = 0 for every p. This completes the proof.

y

x

p

x

n¢n

*
p

Fig. 2.1. An oscillating solution entering the origin (on the left) and its derivative (on the right).

Corollary 2.1:
Let q0 be a singular point of equation (1.2), (1.3). If the coefficients µ0, . . . , µ3 of the cubic
polynomial M do not vanish at q0 simultaneously, there are no oscillating solutions entering
q0. Moreover, in the case of the geodesic equation, d∆(q0) = 0 is a necessary condition for
the existence of an oscillating solution entering q0.

Proof
The first statement is obvious, let us prove the second one.

The geodesic equation for the metric (1.4) has the form (1.2), (1.3), where ∆ = ac− b2

and µ0, . . . , µ3 are expressed through a, b, c by formula (1.5). By Theorem 2.1, for the non-
existence of solutions oscillating at q0 ∈ Γ, it is sufficient to prove that if all coefficients
µ0, . . . , µ3 simultaneously vanish at q0, then d∆(q0) = 0.

Let us assume on the contrary that µi(q0) = 0 for all i and d∆(q0) ̸= 0. To simplify the
calculations, we choose local coordinates centered at q0 ∈ Γ such that b(q0) = 0, which is
always possible to attain by an appropriate linear transformation. Then from the equalities
∆(q0) = 0, d∆(q0) ̸= 0 it follows that |a(q0)|+ |c(q0)| ≠ 0. Without loss of generality one
can assume that a(q0) ̸= 0 and c(q0) = 0. Then from the equalities µi(q0) = 0 and (1.5) it
follows cx(q0) = cy(q0) = 0. Finally, from the equalities

b(q0) = c(q0) = cx(q0) = cy(q0) = 0

it follows d∆(q0) = 0. The obtained contradiction completes the proof.

3. SECOND-ORDER EQUATIONS CUBIC IN THE FIRST-ORDER DERIVATIVE

In this section, we shall briefly describe generic singularities of equation (1.2), whose right-
hand side has the form (1.3). Here we shall assume that ∆, µ0, . . . , µ3 are smooth functions
not connected with each other, in contrast with the geodesic equation, where all these
functions are expressed through a, b, c and their derivatives. According to the genericity

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58 N. G. PAVLOVA, A. O. REMIZOV

assumption, we shall assume that the coefficients µ0, . . . , µ3 of the cubic polynomial (1.3)
do not vanish simultaneously and the set of singular points Γ is a regular curve, which locally
separates the (x, y)-plane into two domains.

Then equation (1.2) has no oscillating solutions, and we consider only proper solutions
passing through or entering a point q0 ∈ Γ. Further we shall omit the word “proper”.

Theorem 3.1:
Solutions of equation (1.2) can enter a point q0 ∈ Γ at directions p that corresponds to the
real roots of M(q0, p) only.

The proof trivially follows from the qualitative analysis of the vector field

ẋ = ∆(q), ẏ = p∆(q), ṗ = M(q, p) (3.15)

generated by equation (1.2) in the 1-jets space.
Given point q0 ∈ Γ, we define the admissible directions p that correspond to singular

points (q0, p) of the vector field (3.15), i.e., satisfy the conditions

∆(q0) = 0, M(q0, p) = 0.

The field (3.15) is a partial case of vector fields

ẋ = v, ẏ = w, żi = aiv + biw, i = 1, . . . , n, (3.16)

where x ∈ R1, y ∈ R1 and z = (z1, . . . , zn) ∈ Rn. Here v, w and ai, bi are smooth functions
on the variables (x, y, z).

The field (3.16) can be invariantly defined as follows: all its components belong to the
ideal I generated by two of them (for instance, v and w). Obviously, the set of singular points
of this field is given by two equations v = 0, w = 0. Therefore, the spectrum of its linear part
at every singular point is

(λ1, λ2, 0, . . . , 0),

where the eigenvalues λ1,2 (either real or complex) continuously depend on the point. If
Reλ1,2 ̸= 0, then the center manifold W c of the field (3.16) coincides with the set of its
singular point, and the Reduction Principle (see [1] or [7]) states that the germ of (3.16) is
topologically equivalent to

ẋ = σ1x, ẏ = σ2y, żi = 0, i = 1, . . . , n,

where σi = sign (Reλi). We recall that the topological equivalence of two vector fields means
that there exists a homeomorphism that sends the phase portrait of the first field to the phase
portrait of the second field.

Smooth local normal forms of vector fields (3.16) are also known; see., e.g. [6] or [24]
(Appendix). Here we present only one result of smooth classification.

Let q0 ∈ Γ and M(q0, p∗) = 0, i.e., p∗ be an admissible direction at q0. The number and
the behaviour of solutions of equation (1.2) that enter q0 with the direction p∗ is determined
by the ratio of the non-zero eigenvalues of the field (3.15):

λ1(q0, p∗) = (∆x + p∆y)(q0, p∗),

λ2(q0, p∗) = Mp(q0, p∗).

Further we shall always assume that λ1,2(q0, p∗) ̸= 0. Define the value

λ = λ2(q0, p∗) : λ1(q0, p∗)

and the set N−1 = {1/n, n ∈ N}.

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OSCILLATING AND PROPER SOLUTIONS 59

Theorem 3.2:
If λ > 0, the germ of (3.16) is C∞-smoothly equivalent to

ξ̇ = a1(ζ)ξ, η̇ = a2(ζ)η, ζ̇j = 0, λ /∈ N ∪ N−1,

ξ̇ = a1(ζ)ξ, η̇ = a2(ζ)η + b2(ζ)ξ
n, ζ̇j = 0, λ ∈ N ∪ N−1\{1},

ξ̇ = a1(ζ)ξ + b1(ζ)η, η̇ = a2(ζ)η + b2(ζ)ξ, ζ̇j = 0, λ = 1.

Using Theorem 3.2, one can get the following result:

Theorem 3.3:
If λ < 0, then equation (1.2) has only one solutions passing though the point q0 with the
tangential direction p∗.

If λ > 0, then equation (1.2) has an infinite number of solutions entering the point q0 with
the direction p∗. In appropriate local coordinates centered at q0, these solutions have one of
two following forms:

y = F (x, c|x|λ), if λ /∈ N,
y = F (x, xλ(c+ ε ln |x|)), ε ∈ {0, 1}, if λ ∈ N,

where F is a smooth function, c = const.

The proof of this theorem is quite similar to those for analogous theorem for the equation
of geodesics in signature varying metrics (see [22] – [19]) although it has a much more
general character. Indeed, for a generic cubic polynomial M(q0, p), there are four (up to
rearrangements of its real roots pi) possible cases:

c1. 1 admissible direction p0 with λ(q0, p0) > 0,
c2. 3 admissible directions p0, p1, p2 with λ(q0, p0) > 0 and λ(q0, pi) < 0 for i = 1, 2,
c3. 1 admissible direction p0 with λ(q0, p0) < 0,
c4. 3 admissible directions p0, p1, p2 with λ(q0, p0) < 0 and λ(q0, pi) > 0 for i = 1, 2.

Here negative λ corresponds to a unique solution passing through q0 with given tangential
direction, while positive λ corresponds to an infinite family of solutions passing through q0
with given tangential direction. For equations of geodesics only the cases c1 and c2 can be
realized.

Example 3.1:
Consider the differential equation

x
dp

dx
= αp(p2 − 1), α ̸= 0, (3.17)

whose singular points fill the curve Γ = {x = 0}. At every q0 ∈ Γ the cubic polynomial

M(p) = αp(p2 − 1)

has three prime roots: p0 = 0 and p1,2 = ±1. Here λ(q0, 0) = −α and λ(q0,±1) = 2α.
Therefore, if α > 0 then equation (3.17) has a single solution passing through the point

q0 with the tangential direction p0 = 0 (the bold line in Fig. 3.2, left) and two infinite families
of solutions entering q0 with the directions p1,2 = ±1 (solid lines in Fig. 3.2, left).

If α < 0 then equation (3.17) has an infinite family of solutions entering the point q0 with
the tangential direction p0 = 0 (solid lines in Fig. 3.2, right) and two single solutions passing
through the point q0 with the directions p1,2 = ±1 (the bold line in Fig. 3.2, right).

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60 N. G. PAVLOVA, A. O. REMIZOV

Fig. 3.2. Solutions of equation (3.17). On the left: α > 0. On the right: α < 0.

Example 3.2:
Consider the differential equation

x
dp

dx
= αp(p2 + 1), α ̸= 0. (3.18)

whose singular points fill the curve Γ = {x = 0}. At every q0 ∈ Γ the cubic polynomial

M(p) = αp(p2 + 1)

has only one real root p0 = 0, whose λ(q0, 0) = α.
Therefore, if α > 0 then equation (3.18) has an infinite family of solutions entering the

point q0 with the tangential direction p0 = 0 (solid lines in Fig. 3.3, left). If α < 0 then
equation (3.18) has a single solution entering the point q0 with the tangential direction p0 = 0
(the bold line in Fig. 3.3, right).

Fig. 3.3. Solutions of equation (3.18). On the left: α > 0. On the right: α < 0.

Example 3.3:
Three more examples taken from [18] present families of geodesics in signature varying
metrics entering a point q0 (or passing through q0) where the metrics degenerates. In the
series [22] – [19], it is shown that for equations of geodesics only the cases c1 and c2 can be
realized.

Three such examples are presented in Fig. 3.4. On the left and center: one admissible
direction p0 = ∞ with λ > 0, an infinite family of geodesics entering q0. On the right: three
admissible directions p0 = ∞ with λ > 0 and p1,2 with λ < 0, which give an infinite family
of geodesics entering q0 with the direction p0 and two single geodesics (bold lines) passing
trough q0 with the directions p1,2.

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OSCILLATING AND PROPER SOLUTIONS 61

Fig. 3.4. Three families of geodesics in degenerate metrics with varying signature.

4. GENERALIZATION FOR HIGHER ORDER EQUATIONS

Here we present some generalizations of the previous results. Consider the quasi-linear
differential equation of the order n ≥ 2 using the notations similar to (2.9):

∆(x, ȳ)
dp

dx
= M(x, ȳ, p), ȳ = (y(0), . . . , y(n−2)), p = y(n−1), (4.19)

where y(i) denotes the i-th derivative of y, and ∆(x, ȳ), M(x, ȳ, p) are smooth functions. It
is important to emphasize that the coefficient ∆ standing with the higher derivative depends
on x, y(0), . . . , y(n−2) and it does not depend on y(n−1). This plays an important role for the
further study.

Denote the locus of singular points of equation (4.19) by Γ i.e.,

Γ = {(x, ȳ) : ∆(x, ȳ) = 0}.

Generically, Γ is a hypersurface in the (x, ȳ)-space. We start with the following definitions
that generalize Definitions 2.1 and 2.2:
Definition 4.1:
Let q0 = (x0, ȳ0) ∈ Γ. A solution y(x) of equation (4.19) enters the point q0 if the following
conditions hold true:

1. The vector-function ȳ(x) is continuous on a non-empty segment Iε with endpoints x0

and x0 + ε, where ε can be either positive or negative, ȳ(x0) = ȳ0.
2. The vector-function ȳ(x) is differentiable and it satisfies (4.19) on the open interval Iε.
3. The graph of ȳ(x) has no common points with Γ except for q0.
If in addition to the above conditions, the function p(x) has a (finite or infinite) limit as

x → x0, the solution y(x) is called proper. Otherwise the solution y(x) is called oscillating
at the point q0.
Definition 4.2:
A solution y(x) is called passing through the point q0 if the vector-function ȳ(x) is
differentiable and it satisfies (4.19) at all points of an open interval I that contains x0,
ȳ(x0) = ȳ0, and the graph of ȳ(x), x ∈ I , intersects the hypersurface Γ at the point q0 only.

Finally, we introduce one more notion. If y(x) is a solution of equation (4.19), the
corresponding vector-function ȳ(x) we shall call an integral curve of this equation.
Theorem 4.1:
Let q0 ∈ Γ and M(q0, p) be an analytic function not identically zero. Then equation (4.19)
has no oscillating solutions entering q0.

The proof of this theorem repeats the proof of Theorem 2.1.

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62 N. G. PAVLOVA, A. O. REMIZOV

Further we shall assume that the conditions of Theorem 4.1 hold true. Moreover, we
assume that Γ is a regular hypersurface, which locally separates the (x, ȳ)-plane into two
domains denoted by D and D′. Then equation (4.19) has no oscillating solutions, and we
shall study its proper solutions (omitting the adjective proper).

Similarly to what was done in Section 3, equation (4.19) generates a direction field in the
(n− 1)-jet space Jn−1 with the coordinates (x, ȳ, p). This direction field is determined by the
vector field

ẋ = ∆(x, ȳ), ẏ(i) = y(i+1)∆(x, ȳ), ẏ(n−2) = p∆(x, ȳ), ṗ = M(x, ȳ, p). (4.20)

The projection π : Jn−1 → Jn−2 (along the p-direction called vertical) of an integral curve of
the field (4.20) different from a straight vertical line, is an integral curve of equation (4.19).

Here we generalize the main statements of Section 3 except for those of them that are
connected with the direction p = ∞. The case p = ∞ is excluded from consideration, since
it cannot be reduced to a finite value similarly to Section 3 and it requires a special detail
research for various types of equations.
Theorem 4.2:
Solutions of equation (4.19) can enter a singular point q0 ∈ Γ only with admissible values of
p that correspond to zeros of the function M(q0, p).

The proof of this theorem repeats the proof of Theorem 3.1.
Definition 4.3:
We shall call a singular point q0 generic, if d∆(q0) ̸= 0, the function M(q0, p) has a finite
number of zeros, all prime, and the corresponding admissible value pi determines the
direction (y(1), . . . , y(n−2), p) in the (x, )̄-space that is transversal to the hypersurface Γ, i.e.,

∆x + y(1)∆y(0) + y(2)∆y(1) + · · ·+ p∆y(n−2) ̸= 0.

Singular points of the field (4.20) have the form (q, pi), where q ∈ Γ and pi is one of zeros
of the function M(q, p). The spectrum of the linear part of the field (4.20) at its singular point
(q0, pi) has the form (0, . . . , 0, λ1, λ2), where

λ1 = ∆x + y(1)∆y(0) + y(2)∆y(1) + · · ·+ p∆y(n−2) , λ2 = Mp

evaluated at (q0, pi). If q0 ∈ Γ is generic, then the both eigenvalues λ1,2 are non-zero. Denote

λ(q0, pi) = λ2(q0, pi)/λ1(q0, pi).

The sign of λ(q0, pi) determines the number of solutions entering the point q0 with the given
admissible direction pi and their rough local properties:
Theorem 4.3:
Let q0 ∈ Γ be a generic singular point of equation (4.19) and pi be an admissible value of p,
that is, M(q0, pi) = 0.

1. If λ < 0, then the equation has a C∞-smooth solution passing through q0 and it has no
other solutions entering q0 with the value pi.

2. If λ > 0, then the equation has an infinite number of solutions entering q0 with the
admissible value pi. Moreover, the family of corresponding integral curves contains an infinite
number of curves entering q0 from the domain D and an infinite number of curves entering
q0 from the domain D′. There exist local coordinates centered at q0 such that solutions
mentioned above have one of two forms:

y = F (x, c|x|λ), if λ /∈ N,
y = F (x, xn(c+ ε ln |x|)), if λ = n ∈ N,

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OSCILLATING AND PROPER SOLUTIONS 63

where ε is 0 or 1, F is a smooth function on two variables, c = const.
Moreover, in the case that equation (4.19) is analytic, in the above statements C∞ can be

replaced with Cω. Then the function F is analytic, and the latter two formulas give Newton–
Puiseux series for solutions.

Theorem 4.3 is similar to Theorem 3.3, the main difference is that now we do not bring
the hypersurface Γ in the (x, ȳ)-space to a simple form (for instance, the hyperplane x = 0).
In general, it is impossible, since the coordinates of the (x, ȳ)-space with n > 2 are not
independent variables. The proof of Theorem 4.3 repeats the proof of Theorem 3.3 with
obvious changes.

5. CONCLUSION

First, we established a necessary condition for quasi-linear equations (1.2) to have an
oscillating solution, which shows that generically such equations have no oscillating solution.
However, the proof of this result (Theorem 2.1) is not appropriate for quasi-linear systems
(1.8), and the question of the existence of oscillating solution for generic systems (1.8) is
still open. The interest to this question is motivated, in particular, by the studying of the
dynamics generated by singular Lagrangians, which goes back to the works of P.A.M. Dirac
and P. G. Bergmann (middle of 20th century).

Second, we described the phenomenon of admissible directions for proper solutions of
equations (1.2), (1.3) and considered the case that such directions correspond prime roots of
the cubic polynomial M . The case of double roots was previously studied in the partial case if
equation (1.2), (1.3) is the geodesic equation generated by a pseudo-Riemannian metric with
varying signature; see the survey [18] or the original works [19, 22, 24]. The study of double
roots for equation (1.2) with a generic cubic polynomial (1.3) is more complicated and it is
not done yet.

ACKNOWLEDGEMENTS

The research is supported by the Russian Foundation for Basic Research (project 20-
01-00610) and by the Ministry of Science and Higher Education of the Russian Federation
(Goszadanie N. 075-00337-20-03, project 0714-2020-0005).

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Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)


	Introduction
	A brief look at quasi-linear equations
	Phenomenon of oscillating solutions
	The aim, scope and structure of the paper

	Oscillating solutions
	Second-order equations cubic in the first-order derivative
	Generalization for higher order equations
	Conclusion