1. Introduction
In recent years, there have been a lot of papers concerning algebraic solutions of first-order autonomous algebraic ordinary differential equations (AODEs), Grasegger [
1] analyzed radical solutions of AODEs. Vo and Zhang [
2] also analyzed rational solutions of AODEs. Feng and Gao [
3,
4] introduced an algorithm to determine the existence of non-trivial rational solutions of autonomous first-order ordinary differential equations, and Winkler [
5] generalized them to systems of autonomous ordinary differential equations. Falkensteiner and Sendra [
6] analyzed formal power series solutions of AODEs. They found that any formal power series solution pertaining to autonomous first-order algebraic ordinary differential equations exhibits convergence. Building upon this finding, Cano, Falkensteiner, and Sendra [
7] broadened the scope to encompass fractional power series solutions and presented an algorithm capable of computing all such solutions. First-order nonlinearly coupled ordinary differential equations were studied in [
8]. Singularities of singular solutions of algebraic ordinary differential equations have also been studied in recent years [
9]. Saji and Takahashi studied singularities of singular solutions of first-order differential equations of the Clairaut type. In [
10], Cui and Hui studied a second-order differential equation with indefinite and repulsive singularities for the first time. Nevertheless, there are merely a handful of steps involved in addressing the challenge of analyzing solutions of autonomous non-algebraic first-order ordinary differential equations. In our research, we focused on studying movable singularities of solutions of autonomous non-algebraic first-order ordinary differential equations in which we consider complex solutions only.
The distinctive features of autonomous non-algebraic first-order ODEs in comparison to other forms of ODEs are described below.
Absence of Explicit Dependence on the Independent Variable: Autonomous ODEs do not explicitly depend on the independent variable. For a first-order autonomous ODE, this means the equation can be written as or in the standard form , where y is the dependent variable and is its derivative. In contrast, non-autonomous ODEs explicitly depend on the independent variable, e.g., .
Invariance to Time Shifts: Autonomous ODEs do not depend on the independent variable; their solutions are invariant to time shifts. This means that if is a solution, then (where c is a constant) is also a solution. This property is not shared by non-autonomous ODEs.
Simplification of the Integration Process: The absence of explicit reliance on the independent variable in autonomous ODEs leads to a streamlining of the integration process. Specifically, as these ODEs do not depend explicitly on variables like time (t), the integration is narrowed down to solely examining the dependent variable and its derivatives. This eliminates the requirement to take into account variations in the independent variable during the integration process, thereby simplifying the steps involved in solving such equations.
Absence of Algebraic Terms: As the name indicates, algebraic ODEs contain algebraic terms or expressions in addition to the derivatives. The algebraic terms can be polynomials, square roots, fractions, etc. Non-algebraic ODEs do not explicitly contain algebraic terms or expressions in their formulation.
Relationship to Dynamical Systems: Solutions of autonomous ODEs can be studied in the phase plane (the
plane), where qualitative features like equilibrium points, stability, and periodic orbits can be easily visualized (see [
11]). Non-autonomous ODEs depend explicitly on time or some other external parameters. This means the system’s behavior evolves not only based on its internal state but also on external influences. Due to this difference, the concept of a cocycle in dynamical systems specifically describes how trajectories of non-autonomous dynamical systems evolve under time shifts. Mathematically, consider a non-autonomous dynamical system governed by an equation of motion
, where
is the state at time
t,
is an external control input, and
f describes the system’s evolution. Cocycle properties describe how the system’s trajectory starting from a point
at time
relates to the trajectory starting from a shifted point
at a later time
. However, since solutions of autonomous ODEs are invariant to time shifts, the concept of a cocycle does not apply to autonomous dynamical systems. For more information on this, especially the stability of non-autonomous systems, please see [
12,
13,
14].
In this paper, we consider autonomous non-algebraic first-order ordinary differential equations of the form
and we also consider autonomous non-algebraic first-order ordinary differential equations of the form
in which
is non-algebraic in
y for
.
In algebraic ordinary differential equations [
15], some singularities are known as movable singularities since their locations change as we transition from one solution to another by altering the initial conditions. For example, the general solution of
is
, where
c is an arbitrary constant. The singularity at
is fixed, while all other singularities (which are located at
,
) are movable square-root branch points. This pertains to a specific type of algebraic singularity where, in the vicinity of such a singularity at
, there exists a rational number
such that the solution can be expressed as the sum of a Laurent series in
with a finite principal part in this region. Readers can find further details in Ince [
16] or Hille [
17]. This is also supported by Painlevé ’s theorem.
Theorem 1. (Painlevé) All movable singularities of all solutions of an equation of the form , where R is rational in y with coefficients that are analytic in z on some common open set, are either poles or algebraic branch points.
For linear equations or linear systems of algebraic ordinary differential equations, singularities are where the coefficients or inhomogeneous terms of the given differential equations or systems become singular. In the case of nonlinear equations or nonlinear systems of algebraic ordinary differential equations, singularities are those points around which the solution function could not be expressed through Taylor expansions or infinite series. For example, consider
In the neighborhood of a singular point
, we can simplify to:
and look for a power-law behavior
. Substituting into Equation (5), we have
and we can substitute
,
to obtain
Obtain the equation for
:
where we can see that
y cannot be expressed as an infinite series around the singular point
. Also, the solution has a pole at
. For examples of solutions of algebraic ordinary differential equations that have algebraic branch points, please see [
15], Theorem 2. The Painlevé theorem concerns the behavior of solutions at movable singular points and ensures that the global behavior of solutions is predictable and controllable. For equations that pass the Painlevé test but whose solutions have more complicated singularities, we can identify base points in the equivalent system of equations, as shown in [
18].
Remark 1. All movable singularities of all solutions of an equation of the form , where I is irrational in y, are either algebraic branch points or logarithmic singularities. For example, the solutions of are ; are algebraic branch points. Additionally, the solutions of are ; are logarithmic singularities. However, since equations are irrational in y, cannot be expressed as a power series of y; intuitively, there are no poles.
If
is algebraic in
y,
implies
, indicating a critical point for autonomous nonlinear equations (see more details on critical points for autonomous nonlinear equations in [
19,
20]). Furthermore, critical points are algebraic singularities, as shown in
Section 2. However, in autonomous non-algebraic first-order ordinary differential equations,
is non-algebraic in
y;
does not necessarily imply
. Readers can gain a more intuitive understanding from the examples in
Section 5. Specifically, we are interested in the behaviors of the singularities in this unstudied case. We aim to prove the main theorems outlined below.
Theorem 2. All movable singularities of all complex solutions of an autonomous first-order ordinary differential equation of the formwhere I is non-algebraic in y, are at most algebraic branch points. We can extend Theorem 2 to the theorem below.
Theorem 3. All movable singularities of all complex solutions of an autonomous first-order ordinary differential equation of the formwhere is non-algebraic in y, are at most algebraic branch points. In
Section 2, we introduce categories of singularities of inverse functions. In
Section 3, we present proofs of Theorems 2 and 3. In
Section 5, we show some examples where all solutions of (1) and (2) have neither non-algebraic nor algebraic singularities, and other examples where all solutions of (1) and (2) exhibit algebraic singularities rather than non-algebraic ones.
The motivation for this study lies in its potential to provide a valuable tool for students and researchers to analyze more singularity problems. The motivation also lies in the potential for innovation and creativity that the constructing triangle method brings; it opens up new possibilities to transform complex solutions of autonomous non-algebraic first-order ODEs into inverse functions of certain trigonometric functions.
2. Categories of Singularities of Inverse Functions
In this section, we introduce several categories of singularities of inverse functions and some lemmas.
Consider
, a non-constant holomorphic map between Riemann surfaces, where
C is the complex plane. Let
be a point in
C such that
. Then, by Theorem 1, there exists a neighborhood
V of the point
and a holomorphic map
such that
. Below, we present the definitions of algebraic and non-algebraic singularities, which are referenced from Alexandre Eremeko and Walter Bergweiler [
21,
22].
Definition 1. (Algebraic Singularity of the Inverse Function) If we define a curve from to , and there is an analytic continuation of ϕ along γ for , for the image . If has a limit point at if , then by continuity , the limit set of must consist of one point; otherwise, the limit set of the curve Γ would contain a continuum, while the preimage of a point under f is discrete. Therefore, Γ ends at . If , then by the inverse function theorem, an inverse function should be defined, which is differentiable through analytic continuation from ϕ to . But in the case of , the inverse function is not defined, so ϕ has an algebraic singularity (branch point) at .
Specifically, if and , or if and is a simple pole of f, then is called an ordinary point. If and , or if and is a multiple pole of f, then is called a critical point and a is called a critical value. Critical points are obvious algebraic singularities.
Definition 2. (Non-algebraic Singularity of the Inverse Function)If we define a curve from to , and there is an analytic continuation of ϕ along γ for , for the image . Suppose extends to ∞, where ∞ is the added point of the complex plane C, making C a Riemann sphere. Γ is a curve parametrized by , as , and has a limit point in G as , so Γ is an asymptotic curve of f. The asymptotic curve of f is the
non-algebraic
singularity of the inverse function of f.
A transcendental singularity is a non-algebraic singularity. There are isolated transcendental singularities and non-isolated transcendental singularities of inverse functions. For detailed definitions, see [
21]. The simplest isolated transcendental singularity of the inverse function is a logarithmic singularity, which is defined below.
Definition 3. (Logarithmic singularity) Given S is an isolated transcendental singularity over a point a, then there is an open disc of radius r around a such that is at a positive distance from other singularities. In the map , does not contain critical values and asymptotic values. is a simply connected region bounded by a simple curve in D, and both ends of the curve tend to ∞. This type of singularity is a logarithmic singularity.
There are two kinds of branch points, logarithmic branch points and algebraic branch points, which are logarithmic singularities and algebraic singularities, respectively. A branch point is a non-isolated singularity.
Definition 4. (Logarithmic branch point)
A logarithmic branch point is a branch point whose neighborhood of values wraps around an infinite number of times as their complex arguments are varied.
Lemma 1. Let . Then is the inverse map on the Riemann surface on which we only choose one piece of one-dimensional manifold that guarantees ’s invertibility. Then, has two logarithmic branch points over ∞ and infinitely many algebraic branch points (critical points) over and 1.
The process of expressing
in the form of a logarithmic function in this proof is referenced from [
23].
Proof. Set
by Euler’s formula. Let
which is equivalent to analyzing the equation
Multiplying by
v on both sides, (11) becomes
Using a basic method to solve quadratic equations, the solution of (11) is
Since
z is a complex variable,
is the complex square-root function. (12) is a multi-valued function with two possible values that differ by a sign,
Since
, it follows that
Since
,
The principal value of the arcsine function is obtained by employing the principal value of the logarithm and the principal value of the square-root function. It is equivalent to employing the principal value of the argument. Therefore, we simplify (13) to
As
is a square-root function,
are branch points.
Substitute
into (14). Then, (14) becomes
with
. Thus,
,
Take
and then
Since
is a branch point of
, it means that
is a branch point of
. Therefore,
has a branch point at
and
. □
Remark 2. If a complex number is represented in polar form , then the logarithm of z is . The logarithm has a jump discontinuity of when crossing the branch point . The singularity of at is a branch point, where is a multi-valued function because θ can be replaced with for any n integer. denotes the principal value of . Obviously, the point under the function is a logarithmic branch point.
4. Methodology
In this section, we provide more detailed illustrations of how to use the constructing triangle method to solve a specific non-algebraic first-order autonomous ordinary differential equation (ODE). This section serves merely as an example of how to apply this method. For other differential equations, readers can be creative and construct other relationships based on the triangle.
Consider a differential equation,
Solve this differential equation:
Assume angle
is located in the second quadrant, as shown in the
Figure 1.
It is easy to see that
where
is a non-negative number.
and square both sides,
Set
and then
By the law of Cosines theorem,
Since
By trigonometric relation,
so we have transformed
into an
function.
Now, we want to further analyze the solutions of (34), which are complex.
In the process of writing (33), we set
, which implies
,
,
,
By substituting (35) into the definition of
, we have
It is apparent that
must be complex in order for
to have a non-zero imaginary part. Since
Squaring both sides, we have
and then
Since is complex, must be complex, which implies . This implies , since .
The next steps are similar to those in
Section 3. Suppose
. Then,
By combining it with (36), we have
We equate the real and imaginary parts:
From
, we have
or
. Take
. Then,
Assume
, and then
Finally, we have the complex angle
From (37), we can easily see that there are no singularities in the solutions of (32) based on the similar analysis in the previous sections.
Comparison with Numerical Methods
When dealing with non-algebraic first-order autonomous ODEs, we may utilize numerical methods. It is essential to check whether non-algebraic first-order autonomous ODEs in the form of (32) are convergent and stable before applying a numerical method, since if the initial value problem is not robust to small perturbations, there is no hope that any numerical method can approximate its solution.
Consider the initial value problem for a system of ODEs
and the perturbed problem
where
is an integrable function and
.
Definition 5. (Stability of the Cauchy problem (see [
11])).
The Cauchy problem (38)
is said to be stable within the time interval if for any perturbations and such thatwe have thatwhere C is a finite constant that does not depend on ϵ. We also want to prove the theorem below.
Theorem 4. Let be an open set, . If is Lipschitz continuous in D and is integrable, then the initial value problem (38) is stable.
Proof. We need to show that for any
and
, the difference between the solutions of (38) and (39) is bounded in some time interval
and that the difference tends to zero as
. First, we notice that if
D is open and
is small enough, then the initial condition
is in
D. If
is Lipschitz continuous,
and since
is integrable, problems (38) and (39) can be equivalently written as
for all
.
Subtracting (43) from (44) and taking the norm yields
and we use Grónwall’s inequality to conclude that
□
This proves that if (38) is Lipschitz continuous, then it is stable. To show that (32) is stable, the remaining task is to show that (32) is Lipschitz continuous.
By the mean value theorem, there exists a
such that
so that
If
I is a trigonometric function in (32), it is easy to see that
. Thus,
is a Lipschitz continuous function in (32). Combined with Theorem 4, (32) is stable.
Since equations in the form of (38) are stable, a numerical scheme can be applied. Here, we use Euler’s method with
to find approximate values for the solutions of the initial value problem (32) at
. The Euler’s scheme is 0-stable and convergent, since
with
and
in such a way that
a constant, then
so the method converges as
.
We rewrite Equation (32) as
Euler’s method yields
By repeating the above process, we eventually obtain the
Figure 2 below.
In the graph, we can easily see that there are no singularities in the solutions of (32).
When comparing the efficiency and accuracy of numerical methods with the constructing triangle method for solving the differential equation , numerical methods may not be as precise as the constructing triangle method. In practice , oscillates as a trigonometric function. By using the constructing triangle method, solutions have been expanded into a logarithmic equation, and the existence of singularities can be easily observed.