First Integrals Without Integrating Factors or Symmetries

Corresponding author: Antonio J. Pan-Collantes, antonio.pan@uca.es

1. Introduction

First integrals are a key concept in the study of ordinary differential equations (ODEs), which represent conserved quantities when the ODE models a physical system. The knowledge of first integrals is an important feature that not only enables the reduction of the order of the ODE, but also facilitates the determination of the general solution, or the validation of numerical methods.

Over the years, numerous techniques have been developed to determine first integrals of ODEs. The most common approaches are based on Lie symmetries. They are extremely successful, but the determination of symmetries can be infeasible for certain classes of ODEs. This limitation has led to the enrichment of the approach through the introduction of more general classes of vector fields, such as dynamical symmetries, nonlocal symmetries, $\mathcal{C}^{\infty}$-symmetries, etc.^{[1][2][3][4][5][6][7][8][9][10]}.

A different point of view to find first integrals is based on the utilization of symbolic computation techniques to determine integrating factors in ODEs. The corresponding integrating factor determining equations, a system of partial differential equations (PDEs), must be solved, and then the first integral can be computed, effectively reducing the order of the ODE. Due to the difficulty of solving these determining equations, various techniques have been developed to aid in the search for integrating factors^{[5][11][12][13][14][15][16][17]}.

While both approaches have significantly advanced the study of first integrals, they come with inherent difficulties, which motivate the research of alternative strategies for finding first integrals. In this paper we present a novel approach, that does not require the use of either symmetries or integrating factors. It is based on the complete integrability of the Pfaffian system associated with the ODE, defined on the corresponding jet bundle, which is the natural framework for the geometric study of ODEs^[18][19].

To provide motivation, consider a first-order ODE of the form 

where $u_1=\frac{du}{dx}$ denotes the derivative of the dependent variable $u$ with respect to the independent variable $x$, and $\phi$ is a smooth function defined on an open set $U\subseteq \mathbb R^2$. Equation (1) is usually written in differential form as

We have, by dimensionality, that $d\theta_{\phi}\wedge \theta_{\phi}= 0$, and therefore, by Frobenius theorem there exists, locally, a smooth function $F=F(x,u)$ such that $dF=\mu\theta_{\phi}$, with $\mu$ a non-vanishing smooth function. The function $F$ can be found by solving the PDE: 

Once a particular solution is found, the general solution to (1) could be expressed in implicit form as 

where $C$ is an arbitrary constant.

In this work, we will generalize this idea up to arbitrary order. After introducing the notation and basic definitions in Section 2, we will present the main result in Section 3. In Section 4 we will particularize the method for the case of second-order ODEs. And in Section 5 we will apply the techniques to third- and fourth-order ODEs.

2. Preliminaries

Consider an $m$th-order ODE 

 where $(x,u)$ represent the independent and dependent variables, respectively, and $u_i$ denotes the $i$th-order derivative of $u$ with respect to $x$, for $1\leq i \leq m$. We consider $(x,u,u_1,\ldots,u_{m-1})$ as the standard coordinates of the corresponding $(m-1)$th-order jet bundle, denoted by $J^{m-1}(\mathbb{R},\mathbb{R})$^[18] for details); and $\phi$ is a smooth function defined on an open subset $U\subseteq J^{m-1}(\mathbb{R},\mathbb{R}).$

Throughout the text, when we say that a fact holds locally, we mean that it is valid in some open subset of $U$. For simplicity, we will continue to use the same notation $U$ to denote such open subsets. Also, we will implicitly use the notation $u_0$ to denote $u$, for the sake of uniformity.

Recall that an important feature of the jet bundle is the contact ideal, which encodes the geometrical structure of this space. The contact ideal is generated by the contact forms, which are 1-forms that vanish on any section of the jet bundle that corresponds to the prolongation of a smooth function $u = f(x)$. The contact forms in the jet bundle $J^N(\mathbb{R}, \mathbb{R})$, $N\geq 1$, are given by: 

Suppose that there exist, locally, two functions $H=H(x,u,\ldots,u_{m-1})$ and $\delta=\delta(x,u,\ldots,u_{m-1})$ such that $\delta$ is non-vanishing and they satisfy 

where $D_x$ denotes the total derivative operator^[1][2] 

The function $H$ is called a first integral of equation (2), and $\delta$ an integrating factor.

The knowledge of a first integral allows us to reduce the order of the ODE (2), since equation (2) is, locally, equivalent to the family of $(m-1)$th-order ODEs 

On the other hand, it is well known^[19] that the ODE (2) is encoded by its associated Pfaffian system, i.e., by the differential ideal $\mathcal I_{\phi}$ of the algebra $\Omega^*(U)$ generated by the contact forms 

together with the 1-form 

This Pfaffian system $\mathcal I_{\phi}$ is in correspondence with the involutive rank-1 distribution generated by the vector field associated to equation (2): 

Therefore, $\mathcal I_{\phi}$ is completely integrable, in the sense that Frobenius theorem applies, i.e., there exist, locally, $m-1$ smooth functions $F_1,\ldots,F_{m-1}$ such that $\mathcal I_{\phi}$ is generated by the 1-forms $dF_1,\ldots,dF_{m-1}$^[19].

Finally, recall that in the case of a single 1-form $\omega$, which corresponds to a corank-1 distribution, Frobenius theorem establishes that the requirement 

is equivalent to the (local) existence of a smooth function $F$ such that $dF=\mu \omega$ for a certain non-vanishing function $\mu$. The 1-form $\omega$ is said to be Frobenius integrable.

3. Main result

Consider an $m$th-order ODE as in (2). Unlike in the first-order case, the 1-form $\theta_{\phi}$ defined in (4) is not, in general, Frobenius integrable. Nevertheless, since the Pfaffian system $\mathcal I_{\phi}$ is completely integrable there must exist, locally, a function $F=F(x,u,\ldots,u_{m-1})$ such that $dF \in \mathcal I_{\phi}$, that is, 

for certain smooth functions $\mu,\alpha_0,\ldots,\alpha_{m-2}$ defined on $U$.

If necessary, we shrink the open set $U$ so that the function $\mu$ is non-vanishing. We then define 

where $\gamma_i:=\frac{\alpha_i}{\mu}$. With this definition, we have 

which implies that $\omega_{(\gamma_0,\ldots,\gamma_{m-2})}$ is Frobenius integrable. Thus, it must satisfy 

which is a first-order PDE system for the functions $\gamma_i$, $0\leq i \leq m-2$.

On the other hand, observe that the smooth function $F$ must satisfy the PDE system 

Now, we are in a position to state and prove the main result of this paper:

Theorem 3.1. Consider an $m$th-order ODE given by (2). A first integral $F$ can be determined by first obtaining a particular solution to the PDE system (6) for the functions $\gamma_i$, and subsequently solving the PDE system (7) for $F$.

Proof. Suppose that the PDE system (6) admits the particular solution 

Then, we use $\gamma_i$ to define the 1-form $\omega_{(\gamma_0,\ldots,\gamma_{m-2})}$ according to equation (5), which is therefore Frobenius integrable.

Consider now a smooth function $F=F(x,u,\ldots,u_{m-1})$ satisfying the PDE system (7). Then, there exists, locally, a certain non-vanishing function $\mu$ such that 

 Expanding both sides of this expression, we have 

 and by comparing the coefficients of the 1-forms $dx,du,du_1,\ldots,du_{m-1}$, we obtain 

So, finally, to check that $F$ is a first integral for the ODE (2) we apply the total differential operator to $F$, and substitute the expressions (8): 

$\square$

Remark 3.1. The PDE system (7) is a homogeneous linear system of first-order PDEs for the function $F$. In contrast, the PDE system (6) is a system of first-order PDEs for the functions $\gamma_i$, which is not generally linear. Moreover, for an $m$th-order ODE, the PDE system (6) consists of $\binom{m+1}{3}$ equations with $m-1$ unknown functions $\gamma_0,\ldots,\gamma_{m-2}$. As a result, solving this system is typically challenging. A practical strategy to address this complexity is to assume some standard ansatz for the functions $\gamma_i$, such as a dependence on fewer variables or linearity in some variable. As we will see in the examples, this kind of assumption not only reduces the complexity of the involved PDEs, but also allows us to write some equations as polynomials, in such a way that they can be split into simpler equations.

Remark 3.2. In certain cases, multiple particular solutions to the PDE system(6) can be identified. Each of these solutions may lead to the construction of distinct, independent first integrals, so increasing the number of conserved quantities or even providing the general solution of the ODE. For an illustration, see Example 4.1.

4. Second-order ODEs

In this section, we will explore the application of our results to the particular case of second-order ODEs. Given the ODE 

 we define the 1-form given by (5) 

where $\gamma_0=\gamma_0(x,u,u_1)$ is a smooth function to be determined.

In this case, condition (6), $d \omega_{(\gamma_0)}\wedge \omega_{(\gamma_0)}=0$, reduces to the single PDE for $\gamma_0$: 

Once a particular solution $\gamma_0 = \gamma_0(x, u, u_1)$ is found, a first integral $F = F(x, u, u_1)$ can be obtained by solving the PDE system (7): 

Remark 4.1. Our approach relates to the framework of the $S$-function method, as introduced in^[20][21]. The $S$-function methodology provides a powerful tool to find first integrals, particularly for rational 2ODEs, by exploiting the relationship between the structure of the equation and associated one-forms. In relation to this work, it can be checked that the function $S$ plays the role of the function $-\gamma_0$.

Remark 4.2. On the other hand, the method presented here relates also to the theory of canonical $\lambda$-symmetries for second-order ODEs. In fact, equation (11) is the determining equation for a canonical $\lambda$-symmetry of the ODE (9), as established in equation (5) in^[22].

The following illustrative example showcases how to use the results above to find first integrals of a second-order ODE.

Example 4.1. Consider the second-order ODE given by: 

To solve the determining equation (11) for $\gamma_0$ we assume the ansatz $\gamma_0=g(x,u)u_1+h(x,u)$, in such a way that (11) simplifies to the polynomial: 

where 

Setting the coefficients $D_0=D_1=D_2=0$ we obtain a system of three PDEs for the functions $g$ and $h$. With the aid of a computer algebra system, we find the particular solution 

and thus we take $\gamma_0=\frac{u_1}{u}$.

By substituting this expression into (12) and clearing the denominators, we obtain the following system of equations: 

 The reader can verify that a particular solution is given by 

which is therefore a first integral of the ODE (13).

Notably, another particular solution to (14) can be checked to be 

and, consequently, the choice $\gamma_0=\frac{3 xu_1-u}{2 x u}$ allows us to find another first integral of (13), provided the corresponding PDE system is solved: 

A particular solution for (17) is given by 

which is, therefore, another first integral of (13).

The first integrals given by (16) and (18) describe the general solutions of the ODE (13) in implicit form.

In the following example, we use our approach to find a first integral and a 1-parameter family of solutions to a second-order ODE that does not admit Lie point symmetries.

Example 4.2. Consider the following second-order ODE 

The reader can check that it does not admit Lie point symmetries, so standard procedures cannot be applied.

To find a first integral we first tackle the PDE (11), which is, in this case, 

By using the natural ansatz $\gamma_0=\gamma_0(x,u_1)$ this equation can be regarded as a polynomial in the $u$ variable: 

 where 

The particular solution $\gamma_0=\frac{1}{u_1}$ for the system given by $D_0=D_1=0$ is easy to find. Now, we write system (12): 

The general solution to equation (21a) is 

where $g=g(x,y)$ is an arbitrary smooth function. From equations (21b) and (21c) it follows that $g$ must satisfy 

This equation admits the particular solution $g(x,y)=y e^{-x^2}$, therefore a particular solution to the system (21) is

which is a first integral of the ODE (19).

Remarkably, even if for the reduced ODE 

no straightforward analytical method yields a general solution, in the particular case $C=0$ the solutions are given by the family 

which in turn is a 1-parameter family of solutions for the ODE (19). Thus, our method has facilitated a partial integration of an ODE for which classical approaches appear to be ineffective.

4.1. Autonomous second-order ODEs

In the case of an autonomous second-order ODE 

the smooth function $\gamma_0=\frac{\phi}{u_1}$ is always a particular solution to equation (11), as can easily be checked. By substituting in (12) the PDE system for the first integral $F=F(x,u,u_1)$ reduces to: 

Observe that the solutions are of the form $F=F(u,u_1)$, so the reduced equation 

 is also autonomous. If $u_1$ can be explicitly isolated in (25) as a function of $u$ (and possibly $C$), the equation can be solved by quadrature, leading to a 2-parameter family of solutions for equation (23).

Example 4.3. Consider the autonomous second-order ODE given by 

which only have the trivial Lie point symmetry $\partial_x$. In this case the PDE system (24), 

admits the particular solution 

Therefore, equation (26) can be reduced to the first-order family of ODEs: 

with $C\in \mathbb R$.

This family of ODEs can be solved by quadrature, leading to the general solution of (26), which is implicitly expressed as: 

where $C,K \in \mathbb R$.

5. Higher-order ODEs

In this section we will show examples of how our approach can be successfully applied to third- and fourth-order ODEs. The general procedure is the same as for second-order ODEs, but the complexity of the PDE systems increases with the order of the ODE.

Example 5.1. Consider the third-order ODE 

According to Theorem 3.1, we need to solve the PDE system (6), 

for the functions $\gamma_0,\gamma_1$. By using the ansatz $\gamma_0=\gamma_0(x,u_2),\gamma_1=\gamma_1(x,u_2)$, the resulting PDE system becomes:

A particular solution can be found by using a computer algebra system, and it is given by 

With these values for $\gamma_0$ and $\gamma_1$, the PDE system (7) for the first integral $F=F(x,u,u_1,u_2)$ becomes 

and a particular solution can be checked to be 

The reduced ODE 

does not appear to admit an explicit closed-form solution in the general case. However, the particular choice $C = 0$ leads to a notable simplification, yielding the 2-parameter family of solutions for (28): 

Observe that the ODE (28) does not possess Lie point symmetries, as the reader may check. Nevertheless, we successfully derived a first integral (31), enabling partial integration of the ODE, without relying on Lie point symmetries or integrating factors.

Example 5.2. Consider the fourth-order ODE

In this case, we have 

and then the 1-form 

 must satisfy condition (6): 

This system of 10 PDEs for $\gamma_0,\gamma_1$ and $\gamma_2$ is too involved to be included in the text. However, by using the ansatz 

we obtain the particular solution 

using a computer algebra system.

Upon substituting these expressions into the PDE system (7), we obtain the following system of equations for the first integral $F$: 

The reader can verify that a particular solution is given by 

Again, we want to point out that our approach has allowed us to find the first integral (35) for the ODE (32) without making use of Lie point symmetries (which are not available for this equation) or integrating factors. Moreover, even if the reduced ODE 

is not easily solved for arbitrary $C\in \mathbb R$, the particular case $C=0$ provides the 3-parameter family of solutions for (32): 

where $K_1,K_2,K_3\in \mathbb R$.

6. Further remarks

In this work, we have presented an approach to find first integrals of $m$th-order ODEs that does not require the computation of integrating factors or the knowledge of Lie point symmetries. Our approach, which is based on the complete integrability of the Pfaffian system associated to the ODE, requires the solution of PDE systems, usually through the assumption of ansätze for the unknown functions and the use of computer algebra systems. The effectiveness of the method has been illustrated with examples of second-, third-, and fourth-order ODEs, showing that it can be applied to a wide range of ODEs, including those that do not admit Lie point symmetries.

We consider that it would be interesting the implementation of our method in computer algebra systems. Such an implementation could be combined with a systematic exploration of different ansätze, for example, by progressively selecting an increasing number of variables for the functions $\gamma_i$ to depend on. This combination could enhance existing algorithms, increasing the likelihood of finding first integrals of ODEs. In this way, our approach offers a complementary tool to the methods currently available, potentially extending the range of solvable problems.