1.Introduction

The exact solutions of nonlinear ordinary differential equations (ODEs) describe the behavior of a great variety of physical, chemical, biological, and engineering systems. Widespread systems in these vast areas of research can be described by homogeneous Liénard equations, which have been intensively studied over the years, see, e.g., ¹ and the recent review ². On the other hand, the same type of inhomogeneous equations received relatively less attention despite the remarkable leap forward brought by the discovery of an irregular noise, later termed deterministic chaos, in the case of sinusoidally driven triode circuits by van der Pol and van der Mark in 1927 ³. Our focus in this short paper is on inhomogeneous Liénard type equations of the form

where the dot denotes the time derivative, d/dt, G(u) and F(u) are arbitrary, but usually polynomial, functions of u, and the forcing term I(t) is an arbitrary continuous function of time.

The main goal of the present paper is to show how the factorization method developed in ^4-6 and the factorization conditions thereof can be used to obtain some integrable inhomogeneous Liénard equations for specific forcing terms. The key point is that the factorization method helps to reduce the inhomogeneous Liénard equations to first-order nonlinear equations, such as Abel and Riccati equations, which are presumably easier to solve in some cases. We recall that the reduction to Riccati equations of the linear Schrödinger equations has been extensively used in supersymmetric quantum mechanics and older factorization methods as reviewed in ^7,8.

2.The nonlinear factorization

As in ^4-6, we consider the factorization of (1)

under the conditions

adding the scheme proposed in ⁹, where one assumes [ddt-f1u]u=Ω(t). This yields the following coupled ODEs for (2),

which we further simplify by taking the second factorizing function as a constant, f2=a2≡const.,

Besides, using the constant function f ₂, conditions (3) and (4) imply a relationship between functions F and G given by

where c ₂ stands for an integration constant, or equivalently

Denoting I(t)=∫0te-a2tI(t)dt, the solution to (7) is

where c₁ is an integration constant given by c1=Ω(0). This allows to rewrite (8) in the form

whose general solution is also the solution of the Liénard Eq. (1), while further particular solutions can be obtained by setting c₁ = 0.

Viceversa, one can say that (12) is a first-order nonlinear reduction of forced Liénard equations of the form

Thus, integrable cases of (12) can provide Liénard solutions in closed form. Since among the most encountered forced Liénard equations are these having F(u) in the form of cubic and quadratic polynomials, in the rest of the paper, we address the applications of this solution method to some cases of these types.

3.The inhomogeneous Duffing-van der Pol oscillator

We choose the particular cubic case F(u)=Au+Cu3 because it corresponds to the forced Duffing-van der Pol oscillator ¹⁰

This equation admits the factorization

where α=A/a2 and γ=C/a2.

The corresponding first-order equation is the Abel equation

The change of variables

turns (16) into the normal form

with invariant

Unfortunately, this formula shows that inhomogeneous Abel equations in this category are not integrable by the separation of variables because N(x) cannot be made constant as required by this type of integrability. Only in the force-free particular case I(t) = 0, the invariant can be reduced to the constant

By separation of variables, the solution is given by the implicit relation

This solution has been obtained previously in ¹⁰.

4.Quadratic inhomogeneous Liénard equations

If we set F(u)=Au+Bu2, then the first order equivalent equation is the Riccati equation

Equation (22) can be transformed into the normal form ¹¹

where

For integrable cases of separable type, one should have N(t) as an arbitrary real constant that we choose p2/4, implying Ω(t)=p2+α2/4β also a constant, as well as a constant driving force

In this simple case, we obtain a Liénard solution of (13) of the form

5.Conclusion

The nonlinear factorization method developed in ^4-6,9 has been used to obtain closed-form solutions of certain types of inhomogeneous Liénard equations. The conditions imposed upon the nonlinear coefficients of the equations by the factorization method and the insertion of the forcing term in the factorization scheme act as designing tools of specific forms of the forcing terms to generate integrable cases by these means. The illustrative examples have been chosen from the class of polynomial (up to cubic) and exponential forcing terms similar to a recent study of inhomogeneous Airy equations ¹². However, the obtained Liénard solutions have rational parts, which make them prone to the presence of singularities. The only regular solutions we have obtained by employing this simple factorization method are the usual tanh kinks. Finally, the scheme presented here is bounded to constant factorization functions f2, since only in this case equation (5) can be turned into the linear (7) in the independent variable t.