In this work, we classify the first integrals of all planar polynomial differential systems of degree one with real constant coefficients. Additionally, we characterize when these first integrals are either polynomial, or rational, or nonalgebraic.

1. Introduction and Statement of the Main Results

Planar polynomial systems are widely studied due to their mathematical tractability and applicability in various fields such as physics, biology, and engineering. One of the fundamental challenges in the qualitative analysis of differential systems is determining the presence of first integrals. The presence of first integrals can greatly facilitate the analysis of these systems by providing valuable information about their dynamics and insights into the geometric and algebraic structure of the system’s solutions. Poincaré attempted to determine whether an algebraic differential equation in two variables has a rational first integral in a series of papers he started in 1891 [1–3]. Many authors have been attracted to the characterization of polynomial or rational integrability for different particular differential systems. For instance, see [4–8] and references therein. In [9], the authors proved that linear differential systems in arbitrary dimension are completely integrable with Darboux first integrals. In our paper, we provide more details than this paper. Recently, the authors in [10] characterized linear differential systems with polynomial first integrals. In the context of a planar polynomial differential system, a “first integral” refers to a nonlocally constant function that remains constant along the trajectories of the system. Specifically, if you have a polynomial differential equation system in two variables and you find a function whose derivative along the trajectories is identically zero, then that function is a first integral of the system. Mathematically, for a planar polynomial differential system given by

()

where P and Q are polynomials, a function H(x, y) is considered as a first integral if dH/ dt = 0 along every solution curve (x(t), y(t)).

We consider polynomial differential systems of degree one,

()

where the dot represents the derivative with respect to time t.

The main result is the following Theorems 1 and 2.

Theorem 1. Depending on real parameters a, b, c, d, e, and f, system (2) has the following global first integral according to the following cases:

[case 1] If 4bd ≠ −(a − e)², and bd − ae ≠ 0, then the first integral of system (2) is given by
()
where . Note that parameter D can be complex.
[case 2] If bd − ae = 0, then e = bd/a. We have the following cases:
- (a)
  If b = 0, e = 0, and a ≠ 0, then the first integral of system (2) is
  ()
- (b)
  If b = 0, e = 0, a = 0, and c ≠ 0, the first integral becomes
  ()
- (c)
  If d = 0, e = 0, f ≠ 0, and a ≠ 0, then the first integral of system (2) is
  ()
- (d)
  If d = 0, e = 0, a = 0, and b ≠ 0, the first integral becomes
  ()
- (e)
  If a ≠ 0, b ≠ 0, and bd ≠ −a², then the first integral of system (2) is given by
  ()
- (f)
  If b = −a²/d and d ≠ 0, therefore the first integral becomes
  ()
[case 3] If 4bd = −(a − e)², we have the following cases:
- (a)
  If b = 0, a = e, and e ≠ 0, then the first integral of system (2) is
  ()
- (b)
  If d = 0, a = e and e ≠ 0, then the first integral of system (2) is
  ()
- (c)
  If c ≠ 0, a ≠ e, and a ≠ −e, then the first integral of system (2) is given by
  ()
- (d)
  If c = 0, a ≠ e, and a ≠ −e, then the first integral of system (2) is given by
  ()
- (e)
  If a = e and e ≠ 0, therefore first integral of (2) is
  ()
- (f)
  If a = −e, therefore the first integral becomes
  ()
- Theorem 1 is proved in Section 2.

By looking at Theorem 1 from an investigative perspective, we can find a rational, polynomial, or nonalgebraic first integral for differential system (2); this is what the following theorem seeks.

Theorem 2. Necessary and sufficient conditions for which system (1) has polynomial, or rational, or nonalgebraic first integrals are one of the following:

[1]
System (2) possesses a polynomial first integral if and only if one of the following conditions is fulfilled:
- (a)
  Condition for [case 1] in Theorem 1, i.e., b ≠ 0, d ≠ 0, d ≠ −((a − e)²/4b), and e ≠ bd/a. In addition, we must have
  ()
- and
  ()
- must be a positive rational. Or a = −e in the case when δ < 0.
- (b)
  Condition for a subcase of [case 1] in Theorem 1, i.e., b = 0, a ≠ 0, e ≠ 0, and a ≠ e. In addition, −a/e must be a positive rational number.
- (c)
  Condition for a subcase of [case 1] in Theorem 1, i.e., d = 0, a ≠ 0, e ≠ 0, and a ≠ e. Additionally, −a/e must be a positive rational number.
- (d)
  Condition for subcase (b) of [case 2] in Theorem 1, i.e., b = 0, a = 0, and c ≠ 0.
- (e)
  Condition for subcase (d) of [case 2] in Theorem 1, i.e., d = 0, e = 0, a = 0, and b ≠ 0.
- (f)
  Condition for subcase (f) of [case 2] in Theorem 1, i.e., e = bd/a, b = −a²/d, c = af/d, and d ≠ 0.
- (g)
  Condition for subcase (f) of [case 3] in Theorem 1, i.e., d = −(a − e)²/(4b) and a = −e.
[2]
System (2) possesses a rational first integral if and only if one of the following conditions is fulfilled:
- (h)
  Condition for [case 1] in Theorem 1, i.e., b ≠ 0, d ≠ 0, d ≠ −((a − e)²/4b), and e ≠ bd/a. In addition, we must have δ > 0, and Δ must be a negative rational.
- (i)
  Condition for a subcase of [case 1] in Theorem 1, i.e., b = 0, a ≠ 0, e ≠ 0, and a ≠ e. In addition, −a/e is a nonnegative rational.
- (j)
  Condition for a subcase of [case 1] in Theorem 1, i.e., d = 0, a ≠ 0, e ≠ 0, and a ≠ e. Additionally, −a/e is a nonnegative rational number.
[3]
A first integral of system (2) is algebraic if and only if one of the conditions [1] or [2] in Theorem 2 holds. Otherwise, it is nonalgebraic.
- Except for the case of a nonalgebraic first integral with δ < 0, which we present in the proof of Theorem 2, Theorem 1 gives the first integral for all cases.
- Theorem 2 is proved in Section 3.

2. Proof of Theorem 1

In most situations, we can apply a linear transformation to the variables so that system (2) becomes either a separable ODE or a unidirectional system (when one of the equations in the planar polynomial system only involves one variable).

Proof of [case 1] in Theorem 1. In these cases, we choose the following affine change of variables

()

where β and γ might be complex parameters. Then, system (2) becomes

()

We nullify both X and Y coefficients in

and

, respectively. With this procedure, system (19) and (20) can be written in the form of the following separable ordinary differential equation:

()

with α₀ = βc + f, α₁ = a/βγ − 1 + a − (bβ/βγ − 1) + (d/β(βγ − 1)) + (d/β) − e/βγ − 1, β₀ = c + γf, and β₁ = −(a/βγ − 1) + (bβ/βγ − 1) − (γd/βγ − 1) + (βγe/βγ − 1). Subject to

()

system (19) and (20) has the first integral

()

Replacing all parameters α₀, α₁, β₁, β₀, β, and γ, therefore system (2) has the first integral (3) under conditions b ≠ 0, d ≠ 0, d ≠ −((a − e)²/4b), and e ≠ bd/a (which are equivalent to α₁ ≠ 0, βγ − 1 ≠ 0, β ≠ 0, β₁ ≠ 0, b ≠ 0, and d ≠ 0).

(i)
If b = 0, we use the linear change of variables (18), with
()
Consequently, system (2) becomes
which is equivalent to the following separable ordinary differential equation:
()
whose solution is
()
The first integral is obtained by solving this equation with respect to the constant k,
()
So the first integral of system (2) in terms of the original variables x, y is given by
()
This expression is equivalent to H₁.
(ii)
If d = 0, the linear change of variables (18) with
()
is used to transform system (2) as follows:

This is equivalent to the following separable ordinary differential equation:

()

which has the solution

()

Solving this equation with respect to the constant k, the first integral is then

()

So the first integral of system (2) in terms of the original variables x, y is given by

()

This expression is also equivalent to H₁.

Proof of [case 2] in Theorem 1. Consider the condition bd − ae = 0, then e = bd/a.

(a)
If b = 0, e = 0, and a ≠ 0, the following separable system is obtained from system (2):
()
The ordinary differential equation dy/ dx = (dx + f)/(ax + c) corresponds to this system and has the following solution:
()
We can obtain the first integral of the system as follows by solving this equation with respect to k, where k is a constant:
()
This leads to equation (4).
(b)
If b = 0, e = 0, a = 0, and c ≠ 0, then the following separable system is obtained from system (2):
()
The ordinary differential equation dy/ dx = (f + dx)/c describes the system and has the following solution:
()
Solving with respect to k, we obtain the first integral of system (2) as follows:
()
Equation (5) is the result of this.
(c)
If d = 0, e = 0, f ≠ 0, and a ≠ 0, using the linear change of variables given by (18) with
()
We can rewrite (2) as
()
This equation is equivalent to the following separable ordinary differential equation:
()
Thus, its solution is
()
and we can obtain the first integral by solving this equation for the constant k:
()
Hence, we can represent the first integral of system (2) in terms of the original variables x, y to obtain (6).
(d)
If d = 0, e = 0, a = 0, and b ≠ 0, then system (2) returns to the following separable differential system:
()
that corresponds to the ordinary differential equation dy/ dx = (f)/(by + c), which has the solution
()
The first integral of the system can be obtained by solving this equation with respect to k, where k is a constant:
()
Equation (7) is obtained as a result of this.
(e)
If a ≠ 0, b ≠ 0, and bd ≠ −a², system (2) is transformed using the linear change of variables (18) with
()
As a result,
()
The following separable ordinary differential equation is equivalent to
()
whose solution is
()
The first integral is obtained by solving this equation for the constant k:
()
Hence, the first integral of (2) using the original variables x and y yields expression (8).
(f)
If b = −a²/d, cd ≠ af, and d ≠ 0, we transform system (2) using the linear change of variables (18) with the values of γ = 0 and β = −(d/a), resulting in the following expressions:
()
leading to an equivalent separable ordinary differential equation

()

whose solution is given by

()

Solving it for the constant k allows us to find the first integral

()

Consequently, the first integration of equation (2) using the original variables x and y results in expression (9).

If c = af/d, system (2) leads to the following expressions:

()

The first integral of the system is also provided by (9).

Proof of [case 3] in Theorem 1. Consider the condition 4bd = −(a − e)², then d = −((a − e)²/4b).

(a)
If b = 0, a = e, and e ≠ 0, system (2) is transformed into the following unidirectional system:
()
which corresponds to the ordinary differential equation dy/ dx = (f + dx + ey)/(c + ex), which has the solution
()
By solving this equation with respect to k, where k is a constant, we obtain the first integral of the system as follows:
()
This results in equation (10).
(b)
If d = 0, a = e, and e ≠ 0, system (2) is converted into the following unidirectional system:
()
This system is equivalent to the ordinary differential equation dy/ dx = (ey + f)/(by + ex + c), whose solution is
()
Solving this equation with respect to k, where k is a constant, yields the first integral of the system as follows:
()
This leads to equation (11).
(c)
If c ≠ 0, a ≠ e, and a ≠ −e, we apply the linear change of variables (18) with the values of β = −(f/c) and γ = (2b/a − e) to effect the transformation of system (2) into the unidirectional form as outlined below:
()
This system is equivalently expressed as the ordinary differential equation
()
and its solution is given by the following equation:
()
Solving this equation with respect to the constant k, the resulting first integral takes the form
()
Consequently, the first integral of system (2) with respect to the original variables x and y results in expression (12).
(d)
If c = 0, a ≠ e, and a ≠ −e, we implement the linear change of variables (18) with the specific values of β = (a − e/2b) and γ = 0 to facilitate the transformation of system (2) into the unidirectional form
()
This system is equivalently portrayed by the ordinary differential equation
()
Its solution is given by
()
When this equation is solved for the constant k, the first integral is as follows:
()
Hence, the first integral of system (2) in terms of the original variables x and y leads to expression (13).
(e)
If a = e and e ≠ 0, the following unidirectional system results from system (2):
()
This system corresponds to the ordinary differential equation dy/ dx = (f + ey)/(by + ex + c), and its solution is given by
()
By solving this equation with respect to k, where k is a constant, we obtain the first integral of the system as follows:
()
This results in equation (14).
(f)
If a = −e, the subsequent system is derived from system (2) as shown in the following equation:
()
This system corresponds to the ordinary differential equation
()
and its solution can be expressed as
()
Upon solving this equation for k, where k is a constant, we derive the first integral of the system
()
leading to the resulting equation (15).

The proof of the first theorem is complete. We will now move on to the second theorem.

3. Proof of Theorem 2

The demonstration of Theorem 2 is mainly based on the preceding Theorem 1 and its proof as well.

Proof of statement [1]-(a) of Theorem 2. We consider conditions of Case 1 of Theorem 1. Since β₁ and α₁ are given in (22), then we obtain α₁ ≠ 0 and β₁ ≠ 0. The right-hand side of equation (23), in terms of (x, y), is in the form

()

If (β₀ + β₁(x + γy)) is a multiple of (α₀ + α₁(βx + y)), with z the factor of multiplication, then by solving the equation

()

we obtain three solutions s₁ = {z = 0, α₀ = 0, α₁ = 0}, s₂ = {α₀ = zβ₀, α₁ = zβ₁, β = 1/γ}, and s₃ = {α₀ = zβ₀, α₁ = 0, β₁ = 0} (since β ≠ 1/γ, see (19) and (20)), which means that one of the factor on the right-hand side of (79) cannot be a multiple of the other.

The algebraicity of H₁ depends on the algebraicity of its factor

as defined in (79). This factor is expressed as

()

In conclusion, the first integral in the case when (16) is satisfied (i.e., δ = (a − e)² + 4bd > 0) is a polynomial if and only if

()

is a positive rational. This completes the proof of statement (1)-(a) for δ > 0.

If δ < 0 and a = −e, then from (82) we obtain −(β₁/α₁) = 1 and in this case, H₁ becomes a polynomial of the form

()

This proves the statement (1)-(a) of Theorem 2 for a = −e and δ < 0.

The case when δ < 0 with a ≠ −e is not discussed here, but it is covered by the proof of statement [3] of Theorem 2.

Proof of statement [1]-(b), [1]-(c), [1]-(d), [1]-(e), [1]-(f), and [1]-(g) of Theorem 2. Theorem 1 implies these statements directly:

(i)
The validity of statement [1]-(b) case becomes evident upon analyzing equation (28) in proof of Theorem 1, assuming that a, e, and a − e are all nonzero.
(ii)
The conclusion of statement [1]-(d) becomes apparent upon examining equation (33) in proof of Theorem 1, under the condition that a, e, and a − e are all distinct from zero.
(iii)
The assertion of statement [1]-(c) can be immediately established by considering equation (5).
(iv)
The substantiation of statement [1]-(e) can be immediately reached by referencing equation (7).
(v)
The proof of statement [1]-(f) becomes evident upon considering equation (9).
(vi)
The establishment of statement [1]-(g) can be immediately accomplished by examining equation (15) in Theorem 1.

Proof of statement [2]-(a) of Theorem 2. It is obvious that the first integral for this case is rational if and only if −β₁/α₁, from equation (82), is a negative rational.

Proof of statement [2]-(b) of Theorem 2. This case is verified by equation (28) in proof of Theorem 1.

Proof of statement [2]-(c) of Theorem 2. This case can be justified by examining (33) in proof of Theorem 1.

Proof of statement [3] of Theorem 2. If δ < 0 and a ≠ −e, then −β₁/α₁ in (82) is not a real number, and since the real and the imaginary parts of

()

from (81) cannot be zero simultaneously or each one separately, then H₁ in Theorem 1 cannot be an algebraic first integral. The first integral in this case can be written in the form (regarding H₁ in Theorem 1)

()

If A is a real number and D = iA (i² = −1), then p = x(a² − ae + 2bd) + y(ab + be) + ac + 2bf − ce, q = aAx + Aby + Ac,

()

and hence, H₁ becomes

()

Then, the real part of H₁ is given as follows:

()

and the imaginary part of H₁ is provided below

()

Then, each function of these two last equations is nonalgebraic.

The following sentences refer to the remaining cases that yield nonalgebraic first integrals:

(i)
Consider the subcase of [case 1] of Theorem 1 for which b = 0. If −a/e is irrational, we arrive at a nonalgebraic first integral due to the conditions a ≠ 0, e ≠ 0, and a ≠ e.
(ii)
Consider the subcase of [case 1] of Theorem 1 for which d = 0. If −a/e is irrational, this leads to a nonalgebraic first integral based on the restrictions a ≠ 0, e ≠ 0, and a ≠ e.
(iii)
All the first integrals in the following cases are evidently nonalgebraic:
- (a)
  Subcase (a) of [case 2] and (a) of [case 3] of Theorem 1.
- (b)
  Subcase (c) of [case 2] and (b) of [case 3] of Theorem 1.
- (c)
  Subcase (e) of [case 2] of Theorem 1.
- (d)
  Subcase (c), (d), and (e) of [case 3] of Theorem 1.

This concludes the entirety of the proof.

Disclosure

This manuscript has been presented as a preprint in Research Square, available at https://www.researchsquare.com/article/rs-3315845/v1.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was supported by the University Mohammed El Bachir El Ibrahimi, Bordj Bou Arreridj, Algerian Ministry of Higher Education and Scientific Research.

Open Research

Data Availability

The data that support the findings of this study are openly available.

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Planar Polynomial Differential Systems of Degree One: Full Characterization of Its First Integrals

Abstract

1. Introduction and Statement of the Main Results

2. Proof of Theorem 1

3. Proof of Theorem 2

Disclosure

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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Planar Polynomial Differential Systems of Degree One: Full Characterization of Its First Integrals

Abstract

1. Introduction and Statement of the Main Results

2. Proof of Theorem 1

3. Proof of Theorem 2

Disclosure

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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