\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conf. 19 (2010),  pp. 123--133.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{123}
\title[\hfilneg EJDE-2010/Conf/19/\hfil First integrals]
{First integrals for the Duffing-van der Pol type oscillator}

\author[G. Gao, Z. Feng \hfil EJDE/Conf/19 \hfilneg]
{Guangyue Gao, Zhaosheng Feng}  % in alphabetical order

\address{Guangyue Gao \newline
Department of Mathematics, University of Texas-Pan American,
Edinburg, TX 78539, USA} 
\email{gygao@broncs.utpa.edu}

\address{Zhaosheng Feng \newline
Department of Mathematics, University of
Texas-Pan American, Edinburg, TX 78539, USA}
\email{zsfeng@utpa.edu fax: 956-381-5091}

\thanks{Published September 25, 2010.}
\thanks{Supported by grant 119100 from UTPA Faculty
Research Council}
\subjclass[2000]{34A25, 34L30}
\keywords{First integral; Duffing oscillator;
 van der Pol oscillator; \hfill\break\indent
Preller-Singer method; parametric solution}

\begin{abstract}
 In this article, under certain parametric conditions,
 we study the first integrals of the Duffing-van der Pol-type
 oscillator equations which include the van der Pol and the
 Duffing oscillator systems, as particular cases.
 After making a series of variable transformations and applying
 the Preller-Singer method, we find the first integrals of the
 simplified equations without complicated calculations.
 Through the inverse transformations we obtain the first
 integrals of the original equations. Some statements in the
 literature are indicated and clarified.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\section{Introduction}

Many nonlinear differential equations arise in physical, chemical
and biological contexts. Finding innovative methods to solve and
analyze these equations has been an interesting subject in the
field of differential equations and \,dynamical systems \cite{jord, pol}.
In these problems, it is not always possible and sometimes not even
advantageous to express exact solutions of nonlinear differential
equations explicitly in terms of elementary functions, but it is
possible to find elementary functions that are constant on
solution curves, that is, elementary first integrals. These first
integrals allow us to occasionally deduce properties that an
explicit solution would not necessarily reveal. In the pioneering
work \cite{prel}, Prelle and Singer introduced a procedure to find
the first integrals of first-order ordinary differential equations (ODEs)
of the form
$y = P(x, y)/Q(x,  y)$, with both $P(x, y)$ and $Q(x, y)$ polynomials whose
coefficients lie in the field of complex numbers $\mathbb{C}$.
Duarte et al. \cite{dua} extended this procedure to second-order
ODEs which is based on a conjecture that if the given second-order
ODE has an elementary solution, then there exists at least one
elementary first integral $I(x, y, y')$ whose derivatives are
all rational functions of $x$, $y$ and $y'$. Recently,
much attention has been received to various nonlinear oscillator systems for
finding the first integrals and obtaining exact
solutions \cite{cha1,cha3, jord, sen}. Special types of first
integrals and exact solutions are of
fundamental importance to our understanding of physical, chemical
and biological phenomena modelled differential equations.


In this article, we consider a more general nonlinear oscillator system
of the form
\begin{equation}\label{ee1}
    \ddot{u}+(\delta+\beta{u^{m}}){\dot{u}}-\mu{u}+\alpha{u}^{n}=0.
\end{equation}
where an over-dot represents differentiation with respect to the
independent variable $\xi$, and all coefficients
$\delta, \beta, \mu$ and $\alpha$ are real. It is referred as to
the Duffing--van der Pol--type oscillator, since the choices
$\alpha = 0$ and $m = 2$ lead equation \eqref{ee1} to the van der
Pol oscillator
\begin{equation}\label{ee2}
\ddot{u}+(\delta+\beta{u^{2}}){\dot{u}}-\mu{u}=0,
\end{equation}
which was originally discovered by the Dutch electrical engineer
van der Pol in electrical circuits \cite{van1, van2}. The choices
$\beta = 0$ and $n=3$ lead equation \eqref{ee1} to the damped
Duffing equation \cite{duff1, guc}
\begin{equation}\label{ee3}
\ddot{u}+\delta \dot{u}-\mu{u}+\alpha{u}^{3}=0.
\end{equation}
When $\beta=0$ and $n=2$, equation \eqref{ee1} becomes the damped
Helmholtz oscillator \cite{alm, ras}
\begin{equation}\label{ee4}
\ddot{u}+\delta \dot{u}-\mu{u}+\alpha{u}^{2}=0.
\end{equation}
It is well known that there are a great number of theoretical
works to deal with equations \eqref{ee2}--\eqref{ee4} \cite{guc,
jord}, and applications of these three equations and the related
equations can be seen in quite a few scientific areas \cite{git}.


In the present paper, we wish to show that under certain
parametric conditions some first integrals of oscillator system
\eqref{ee1} are established. The paper is organized as follows. In
the next section, in order to make this paper well self-contained,
we summarize the Prelle--Singer procedure developed by Duarte et
al. \cite{dua} for constructing the first integrals of
second-order ODEs. In Section 3, we will show that after a series
of nonlinear transformations, we simplify equation \eqref{ee1},
then by means of the Preller--Singer method we derive the first
integral of the simplified equation without complicated
calculations. Through the inverse transformations we obtain the
first integral of the original oscillator equation. Some
statements in the literature are indicated and clarified, and some
exact solutions of equation \eqref{ee1} in the parametric forms
are obtained accordingly. Section 4 is a brief conclusion.


\section{Prelle--Singer Method for Solving Second-Order ODEs}

In this section, in order to present our results in a
straightforward way, we start our attention by briefly reviewing the
Prelle--Singer procedure for solving second-order ODEs developed
by Duarte et al. \cite{dua} and Chandrasekar et al. \cite{cha1}.

Consider the second-order ODE of the rational form
\begin{equation}\label{e1}
\frac{d^{2}
y}{\,dx^{2}}=\phi(x, y, y')=\frac{P(x, y, y')}{Q(x, y, y')},\quad
P, Q \in \mathbb{C}[x,  y,  y'].
\end{equation}
where $y'$ denotes differentiation with respect to $x$, $P$ and
$Q$ are polynomials in $x,  y$ and $y'$ with coefficients in the
complex field. Suppose that equation \eqref{e1} admits a first
integral $I(x,  y,  y')=C$, with $C$ constant on the solutions,
so we have the total differential
\begin{equation}\label{e2}
dI=I_{x}\,dx+I_{y}\,dy+I_{y'}\,dy'=0,
\end{equation}
where the subscript denotes partial differentiation with respect
to the corresponding variable. On the solution, since $y' \,dx=\,dy$
and equation \eqref{e1} is equivalent to $\frac{P}{Q}\,dx=\,dy'$,
adding a null term $S(x, y, y')y' \,dx-S(x, y, y') \,dy$ to both
sides yields
\begin{equation}\label{e4}
\big( \frac{P}{Q}+Sy' \big)\,dx-S\,dy-\,dy'=0.
\end{equation}
 From \eqref{e2} and \eqref{e4}, one can see that on the solutions,
the corresponding coefficients of \eqref{e2} and \eqref{e4} should
be proportional. There exists a proper integrating factor
$R(x, y, y')$ for expression \eqref{e4}, such that on the
solutions
\begin{equation}\label{e5}
dI=R(\phi+Sy')\,dx-SR\,dy-R\,dy'=0.
\end{equation}
Comparing the corresponding terms in \eqref{e2} and \eqref{e5}, we
have
\begin{equation} \label{e6}
\begin{gathered}
I_{x}=R(\phi+Sy'), \\
I_{y}=-SR, \\
I_{y'}=-R,
\end{gathered}
\end{equation}
and the compatibility conditions $I_{xy} = I_{yx}$,
$I_{xy'} =I_{y'x}$ and $I_{yy'} = I_{y'y}$. Using these three
compatibility conditions respectively, we obtain three equivalent
equations as follows:
\begin{equation} \label{e7}
\begin{gathered}
D[S]=-\phi_{y}+S\phi_{y'}+S^{2}, \\
D[R]=-R(S+\phi_{y'}), \\
R_{y}=R_{y'}S+S_{y'}R,
\end{gathered}
\end{equation}
where $D$ is an differential operator
\[
D = \frac{\partial}{\partial x}+y'\frac{\partial}{\partial
y}+\phi\frac{\partial}{\partial y'}.
\]
For the given expression of $\phi$, one can solve the first
equation of \eqref{e7} for $S$. Substituting $S$ into the second
equation of \eqref{e7} one can get an explicit form for $R$ by
solving it. Once a compatible solution $R$ and $S$ satisfying the
extra constraint (the third equation of \eqref{e7}) is derived,
integrating \eqref{e6}, from \eqref{e2} one may obtain a first
integral of motion as follows
\begin{equation} \label{ee8}
\begin{aligned}
&I(x, y, y')\\
&=\int R(\phi+Sy')\,dx-\int \Big[ RS+\frac{\partial}
{\partial y}\int R(\phi+Sy')\,dx \Big]\,dy  \\
 &\quad -\int \Big\{ R+\frac{\partial}{\partial y'} \Big( \int
R(\phi+Sy')\,dx-\int \Big[ RS+\frac{\partial}{\partial y}\int
R(\phi+Sy')\,dx \Big] \,dy \Big)\Big\}\,dy'.
\end{aligned}
\end{equation}

\section{First Integrals of Nonlinear Oscillator Systems}

\subsection{Nonlinear Transformations}

In this subsection, in order to avoid doing complicated
computations, we will make a series of nonlinear transformations
to equation \eqref{ee1}. For our convenience, we assume $\alpha=1$
in equation \eqref{ee1} (this can be easily obtained by re-scaling
parameters of equation \eqref{ee1}). Namely, we consider the
oscillator equation:
\begin{equation}\label{e8}
    \ddot{u}+(\delta+\beta{u^{m}}){\dot{u}}-\mu{u}+{u}^{n}=0.
\end{equation}
Firstly, we make the natural logarithm transformation:
\begin{equation}\label{ec8}
    \xi=-{\frac{1}{\delta}}{\ln\tau};
\end{equation}
that is,
\[
    \frac{\partial\tau}{\partial\xi}=-\delta{e^{-\xi\delta}}=-\delta\tau.
\]
After substituting the following two derivatives into
\eqref{e8}:
\begin{gather*}
   \frac{\partial{u}}{\partial\xi}=\frac{\partial{u}}{\partial\tau}*
  \frac{\partial\tau}{\partial\xi}=-\delta\tau\frac{\partial{u}}{\partial\tau}, \\
   \frac{\partial^{2}u}{\partial\xi^{2}}=\delta^{2}\tau\frac{\partial{u}}
  {\partial\tau}+\delta^{2}\tau^{2}\frac{\partial^{2}u}{\partial\tau^{2}},
\end{gather*}
then it becomes
\begin{equation}\label{e9}
    \delta^{2}\tau^{2}{\frac{\partial^{2}{u}}{\partial\tau^{2}}}
    -\beta\delta\tau{u^{m}}{\frac{\partial{u}}{\partial{\tau}}}-\mu{u}+u^{n}=0.
\end{equation}
Further, we take the variable transformation as:
\begin{equation}\label{ee9}
    q=\tau^{\kappa}, \quad u=\tau^{-\frac{1}{2}(\kappa-1)}H(q),
\end{equation}
A direction calculation gives
\begin{gather*}
   \frac{\partial{u}}{\partial\tau}
  =-\frac{1}{2}(\kappa-1)q^{-\frac{\kappa+1}{2\kappa}}H(q)
  +\kappa{q^{\frac{\kappa-1}{2\kappa}}}\frac{\partial{H}}{\partial{q}},\\
 \frac{\partial^{2}u}{\partial\tau^{2}}=\frac{1}{4}(\kappa^{2}-1)q^{-\frac{\kappa+3}{2\kappa}}H(q)
  +\kappa^{2}q^{\frac{3(\kappa-1)}{2\kappa}}\frac{\partial^{2}H}{\partial{q^{2}}}.
\end{gather*}
After substituting the above equalities into  \eqref{e9},
we obtain
\begin{equation}\label{e10}
    \frac{\partial^{2}{H}}{\partial{q}^{2}}=\frac{\beta}{\delta\kappa}q^{\frac{m-\kappa(m+2)}{2\kappa}}
    H^{m}\frac{\partial{H}}{\partial{q}}-\frac{1}
    {\delta^{2}\kappa^{2}}q^{\frac{-(3+n)\kappa+n-1}{2\kappa}}H^{n}
-\frac{1}{2}\frac{(\kappa-1)\beta}{\delta\kappa^{2}}H^{m+1}
q^{\frac{m-\kappa(m+4)}{2\kappa}},
\end{equation}
where an over-dot represents differentiation with respect to the
independent variable $q$, and
\begin{equation}\label{ee10}
    \kappa^2= \frac{4\mu}{\delta^{2}}+1.
\end{equation}

\subsection{Force-Free Duffing-van der Pol Oscillator}

We know that the choices $m = 2$ and $n = 3$ lead equation
 \eqref{ee1} to the standard form of the Duffing-van der Pol
oscillator equation,
whose autonomous version (force-free) is:
\begin{equation}\label{e12}
    \ddot{u}+(\delta+\beta{u^{2}})\dot{u}-\mu {u}+u^{3}=0.
\end{equation}
Equation \eqref{e12} arises in a model describing the propagation
of voltage pulses along a neuronal
axon and has recently received much attention from many authors.
A vast amount of literature exists
on this equation; for details and applications,
see \cite{hol, lak} and references therein.

 From  \eqref{e10}, one can see that if we take $n=3$ and
$m=2$, then equation \eqref{e10} can be reduced to a simple form
\begin{equation}\label{ee12}
    \frac{\partial^{2}H}{\partial{q}^{2}}=Aq^{p}H^{2}
    \frac{\partial{H}}{\partial{q}}+B q^{p-1}H^{3},
\end{equation}
where
\begin{gather*}
   p=\frac{1}{\kappa}-2,\quad A=\frac{\beta}{\delta\kappa}, \\
   B=-\frac{1}{\delta^{2}\kappa^{2}}
-\frac{(\kappa-1)\beta}{2\delta\kappa^{2}}.
\end{gather*}
Choosing $\phi(q, H, H')=Aq^{p}H^{2}
    \frac{\partial{H}}{\partial{q}}+B q^{p-1} H^{3}$
and following the procedure in Section 2, we obtain three
determining equations:
\begin{gather}
  S_{q}+\dot{H}S_{H}+\phi{S_{\dot{H}}}=-2Aq^{p}H\dot{H}
 +(ASq^{p}-3Bq^{p-1})H^{2}+S^{2}, \label{ee13} \\
  R_{q}+R_{H}\dot{H}+\phi{R_{\dot{H}}}=-RS-RAq^{p}H^{2}, \label{ee14} \\
  R_{H}=R_{\dot{H}}S+S_{\dot{H}}R. \label{ee15}
\end{gather}
In general, it is not easy to solve system \eqref{e7} and get
exact solutions ($S, R$) in the explicit forms. But in our case
of \eqref{ee13}-\eqref{ee15} we may seek an ansatz for $S$ and $R$
of the forms as suggested in \cite{dua}:
\begin{equation}\label{e16}
    S=\frac{a(q,H)+b(q,H)\dot{H}}{c(q,H)+d(q,H)\dot{H}},   \quad
R=e(q,H)+f(q,H)\dot{H},
\end{equation}
where $a,  b,  c,  d, e,  f$ are functions of $q,  H$ to
be determined. Substituting $S$ into equation \eqref{ee13}, we get
the equation system
\begin{eqnarray*}
   && [\dot{H}]^{0}: -3Bc^{2}H^{2}q^{p-1}+Aacq^{p}H^{2}+a^{2}
   =a_{q}c-ac_{q}+bcBH^{3}q^{p-1}-adBH^{3}q^{p-1}, \\
   && [\dot{H}]^{1}:
   -2Ac^{2}q^{p}H-6BcdH^{2}q^{p-1}+2Aadq^{p}H^{2}+2ab\\
   && \hspace{0.4in}=a_{q}d+b_{q}c-ad_{q}-bc_{q}+a_{H}c-ac_{H},\\
   && [\dot{H}]^{2}:
   -4Acdq^{p}H-3Bd^{2}H^{2}q^{p-1}+Abdq^{p}H^{2}+b^{2}\\
   && \hspace{0.4in}=b_{q}d-bd_{q}+a_{H}d+b_{H}c-ad_{H}-bc_{H}, \\
   && [\dot{H}]^{3}: -2Ad^{2}q^{p}H=b_{H}d-bd_{H}.
\end{eqnarray*}
Substituting $S$ and $R$ into equation \eqref{ee14}, we obtain
another equation system:
\begin{gather*}
   [\dot{H}]^{0}: e_{q}c+BcfH^{3}q^{p-1}=-ae-Aceq^{p}H^{2}, \\
   [\dot{H}]^{1}: f_{q}c+e_{H}c+2Afcq^{p}H^{2}+e_{q}d+BfdH^{3}q^{p-1}=-be-Adeq^{p}H^{2}-af,\\
   [\dot{H}]^{2}: f_{H}c+f_{q}d+e_{H}d+2Afdq^{p}H^{2}=-bf,\\
   [\dot{H}]^{3}: f_{H}d=0.
\end{gather*}
Under the parametric condition
\begin{equation}\label{e19}
    \delta=\frac{3}{\beta}-\frac{\mu \beta}{3},
\end{equation}
we solve the above two nonlinear systems for a nontrivial solution
with the aid of Maple, and the corresponding forms of $S$ and $R$
read
\begin{equation}\label{e17}
    S=-\frac{1}{q}-\frac{\beta}{\delta\kappa}
q^{\frac{1-2\kappa}{\kappa}}H^{2}, \quad
    R=e^{\ln{q}},
\end{equation}
which also satisfy equation \eqref{ee15}.

Substituting the solution set \eqref{e17} into
\eqref{ee8}, we can obtain the first integral of equation
\eqref{ee12} immediately:
\begin{equation}\label{e18}
   \kappa\delta{H}-\kappa\delta{q\dot{H}}+\frac{2}{\delta(1-\kappa)}q^{\frac{(1-\kappa)}{\kappa}}H^{3}=I.
\end{equation}
Using the inverse transformations \eqref{ec8} and \eqref{ee9}, and
changing to the original variables, we obtain that under the
parametric condition \eqref{e19}, the Duffing-van der Pol equation
\eqref{e12} has the first integral of the form
\begin{equation}\label{e22}
    \left[ \dot{u}+\left(\delta-\frac{3}{\beta} \right)u+\frac{\beta}{3}u^{3} \right]
    e^{\frac{3 \xi}{\beta}}=I_1.
\end{equation}


It is remarkable that in \cite[p. 2467]{cha1},  \cite[p.4528]{cha3},
and \cite[p. 1936]{sen}, the authors studied the first
integral of the oscillator equation \eqref{e12} by the Lie
symmetry method  and claimed that the nontrivial first
integral exists only for the parametric choice
\begin{equation}\label{3ee10}
\delta=\frac{4}{\beta},\quad \mu=-\frac{3}{\beta^2}.
\end{equation}
However, in view of our condition \eqref{e19} and formula
\eqref{e22}, it shows that our parametric constraint \eqref{e19}
is weaker than the corresponding ones described in the literature
\cite{cha1, cha3, sen}, and the first integral presented in
\cite{cha1, cha3, sen} is just a particular case of \eqref{e22}.

\subsection{Duffing--van der Pol--Type Oscillator}

In this subsection, we extend the technique used in the preceding
subsection to a more general oscillator equation in the case of
$n=m+1$; that is,
\begin{equation}\label{e23}
    \ddot{u}+(\delta+\beta{u^{m}})\dot{u}-\mu{u}+u^{m+1}=0,
\end{equation}
where an over-dot still denotes differentiation with respect to
$\xi$. Note that the choice $n=m+1$ leads equation \eqref{e10} to
a simple form
\begin{equation}\label{e24}
    \frac{\partial^{2}H}{\partial{q}^{2}}=\frac{\beta}{\delta\kappa}q^{p}H^{m}
    \frac{\partial{H}}{\partial{q}}+\left(-\frac{1}{\delta^{2}\kappa^{2}}-\frac{(\kappa-1)
    \beta}{2\delta\kappa^{2}}\right) H^{m+1}q^{p-1},
\end{equation}
where
\[ %\label{e25}
   p=\frac{m-\kappa(m+2)}{2\kappa}.
\]
For the notational convenience, we denote that
\[ %\label{e26}
    A=\frac{\beta}{\delta\kappa}, \quad
B=-\frac{1}{\delta^{2}\kappa^{2}}
-\frac{(\kappa-1)\beta}{2\delta\kappa^{2}},
\]
then equation \eqref{e24} becomes
\begin{equation}\label{e27}
    \ddot{H}=Aq^{p}H^{m}\dot{H}+BH^{m+1}q^{p-1}.
\end{equation}

Choosing $\phi(q, H, H')=Aq^{p}H^{m}
    \frac{\partial{H}}{\partial{q}}+B q^{p-1} H^{m+1}$
and following the procedure in Section 2, we obtain three
determining equations:
\begin{gather}
  S_{q}+\dot{H}S_{H}+\phi{S_{\dot{H}}}=-mAq^{p}H^{m-1}\dot{H}
 +(ASq^{p}-(m+1)Bq^{p-1})H^{m}+S^{2}, \label{3.3ee1} \\
  R_{q}+R_{H}\dot{H}+\phi{R_{\dot{H}}}=-RS-RAq^{p}H^{m}, \label{3.3ee2} \\
  R_{H}=R_{\dot{H}}S+S_{\dot{H}}R. \label{3.3ee3}
\end{gather}
Here we use the same ansatz for $S$ and $R$ as given in
\eqref{e16}. Substituting $S$ into  \eqref{3.3ee1}, we get
the  system
\begin{eqnarray}
   && [\dot{H}]^{0}:
   -(m+1)Bc^{2}H^{m}q^{p-1}+Aacq^{p}H^{m}+a^{2} \nonumber\\
   && \hspace{0.4in}=a_{q}c-ac_{q}+bcBH^{m+1}q^{p-1}-adBH^{m+1}q^{p-1}, \nonumber\\
   && [\dot{H}]^{1}: -mAc^{2}q^{p}H^{m-1}-2(m+1)BcdH^{m}q^{p-1}+Aadq^{p}H^{m}+Aq^{p}H^{m}bc+2ab\nonumber\\
   && \hspace{0.4in}=a_{q}d+b_{q}c-ad_{q}-bc_{q}+a_{H}c-ac_{H}+bcAq^{p}H^{m}-adAq^{p}H^{m},\nonumber\\
   && [\dot{H}]^{2}: -2mAcdq^{p}H^{m-1}-(m+1)Bd^{2}H^{m}q^{p-1}+Abdq^{p}H^{m}+b^{2} \label{3ee1}\\
   && \hspace{0.4in}=b_{q}d-bd_{q}+a_{H}d+b_{H}c-ad_{H}-bc_{H}, \nonumber\\
   && [\dot{H}]^{3}: -mAd^{2}q^{p}H^{m-1}=b_{H}d-bd_{H}.\nonumber
\end{eqnarray}
Substituting $S$ and $R$ into \eqref{3.3ee2}, we obtain another
system,
\begin{equation}
\begin{gathered}{}
[\dot{H}]^{0}: e_{q}c+BcfH^{m+1}q^{p-1}=-ae-Aceq^{p}H^{m}, \\
[\dot{H}]^{1}: f_{q}c+e_{H}c+2Afcq^{p}H^{m}+e_{q}d+BfdH^{m+1}q^{p-1}
  =-be-Adeq^{p}H^{m}-af,\\
[\dot{H}]^{2}: f_{H}c+f_{q}d+e_{H}d+2Afdq^{p}H^{m}=-bf,\\
 [\dot{H}]^{3}: f_{H}d=0.
\end{gathered} \label{3ee2}
\end{equation}
We solve the nonlinear systems \eqref{3ee1} and
\eqref{3ee2}, for a nontrivial solution, with the aid of Maple and
find that under the parametric conditions
\begin{equation}
\begin{gathered}\label{e33}
     m=\frac{(1-\kappa)\beta\delta}{2}-1,
    \quad \kappa^2=\frac{4\mu}{\delta^{2}}+1,
\end{gathered}
\end{equation}
the three determining equations \eqref{3.3ee1}--\eqref{3.3ee3}
have the solution of the form
\begin{equation}\label{3ee3}
    S=-\frac{1}{q}-\frac{\beta}{\delta\kappa}
q^{\frac{m(1-\kappa)}{2\kappa}-1}H^{m},  \quad
    R=e^{\ln{q}}.
\end{equation}

After substitution of the solution set \eqref{3ee3} into
\eqref{ee8}, we derive the first integral of  \eqref{e27}
as follows
\[
   \kappa\delta{H}-\kappa\delta{q\dot{H}}
+\frac{2}{\delta(1-\kappa)}q^{\frac{m(1-\kappa)}{2\kappa}}H^{m+1}=I,
\]
where $I$ is an arbitrary integration constant. By virtue of the
inverse transformations \eqref{ec8} and \eqref{ee9}, and changing to
the original variables, we obtain that under the parametric
condition \eqref{e33}, the Duffing-van der Pol--type equation
\eqref{e23} has the first integral of the form
\begin{equation}\label{e42}
    \left[\dot{u}+\frac{\delta(\kappa+1)}{2}u+\frac{2}{\delta(1-\kappa)}
    u^{m+1}\right]e^{\frac{1}{2}\delta(1-\kappa)\xi}=I_2.
\end{equation}

It is remarkable that the first integral of the Duffing-van der
Pol oscillator equation \eqref{e12} obtained in Section 3.2 is
just a particular case of formula \eqref{e42}. In the recently
published Handbooks of ODEs such as \cite{can, pol,zai}, there are
quite a few first integrals (conservation laws) collected for
ordinary differential  equations of the type
$y''=c_1x^{l_1}y^{m_1}(y')^{k_1}+c_1x^{l_2}y^{m_2}(y')^{k_2}$, but
our formulas of first integrals of equation \eqref{e23} or
\eqref{e24} described herein are not presented there.


\subsection{Solutions in the Parametric Forms}

In this subsection, by virtue of the first integral \eqref{e42},
we may choose a proper value for $I_2$ and consider three
particular cases where exact solutions of the oscillator equation
\eqref{e23} can be expressed in the parametric forms.


\noindent {\bf Case 1:} Assume that $m \neq -1$ and $\kappa \neq -1$,
and
\begin{equation} \label{e45}
\begin{gathered}
   m=-\frac{2\kappa}{\kappa+1},\\
   \frac{\beta}{\delta\kappa}=\frac{1}{\delta^{2}\kappa^{2}}
    +\frac{(\kappa-1)\beta}{2\delta\kappa^{2}},
\end{gathered}
\end{equation}
where
\[
\kappa^2=\frac{4\mu}{\delta^{2}}+1.
\]
In this case,  \eqref{e24} takes the form
\begin{equation}\label{e43}
    \ddot{H}=Aq^{-m-2}H^{m}\dot{H}-AH^{m+1}q^{-m-3}.
\end{equation}
 From the first integral \eqref{e42}, taking $I_2=0$, we know that
the solution of equation \eqref{e43} can be expressed in the
parametric form \cite{pol}:
\begin{equation} \label{e44}
\begin{gathered}
  q=aC_{1}^{m}\Big(\int\frac{dt}{1\pm{t^{m+1}}}+C_{2}\Big)^{-1}, \\
  H=bC_{1}^{m+1}t \Big(\int\frac{dt}{1\pm{t^{m+1}}}+C_{2}
\Big)^{-1},
\end{gathered}
\end{equation}
where $C_1$ and $C_2$ are arbitrary constants, $a$ and $b$ are
also arbitrary but satisfy
\begin{equation}\label{4ee1}
    \frac{\beta}{\delta\kappa}=\mp(m+1)a^{m+1}b^{-m}.
\end{equation}
Applying the inverse transformation of \eqref{ee9} to formula
\eqref{e44}, namely
\[
    \tau=q^{\frac{1}{\kappa}}, \quad
H=u\tau^{\frac{1}{2}(\kappa-1)},
\]
we have
\begin{equation}
\begin{gathered}
  \tau=a^{\frac{1}{\kappa}}C_{1}^{m/\kappa}
  \Big( \int\frac{dt}{1\pm{t^{m+1}}}+C_{2} \Big)^{-1/\kappa}, \\
  u=\tau^{-(\kappa-1)/2}bC_{1}^{m+1}t
  \Big(\int\frac{dt}{1\pm{t^{m+1}}}+C_{2}\Big)^{-1}.
\end{gathered} \label{e50}
\end{equation}
Further, applying the inverse transformation of \eqref{ec8} to
formula \eqref{e50}, under the given parametric condition
\eqref{e45}, we obtain the solution for equation \eqref{e23} in
the parametric form as follows:
\begin{equation}
\begin{gathered}
 \xi=\frac{-\ln\Big(a^{\frac{1}{\kappa}}C_{1}^{m/\kappa}
\Big(\int\frac{dt}
  {1\pm{t^{m+1}}}+C_{2}\Big)^{-1/\kappa}\Big)}{\delta}, \\
 u=e^{\delta (\kappa-1)/2\xi}bC_{1}^{m+1}t
  \Big(\int\frac{dt}{1\pm{t^{m+1}}}+C_{2} \Big)^{-1},
\end{gathered} \label{e52}
\end{equation}
where $a$ and $b$ are arbitrary constants, and satisfy condition
\eqref{4ee1}.



\noindent{\bf Case 2:} Assume that
\begin{equation}\label{3.4e1}
    m=-2, \quad \kappa=-2, \quad \beta\delta=-2.
\end{equation}
So  \eqref{e24} takes the form
\begin{equation}\label{3.4e2}
    \ddot{H}=Aq^{1/2}H^{-2}\dot{H}-AH^{-1}q^{-1/2},
\end{equation}
where $A=-\beta/(2 \delta)$.

Using the first integral \eqref{e42} again, we know that the
solution of equation \eqref{3.4e2} can be expressed in the
parametric form:
\begin{equation}
\begin{gathered}
   q=aC_{1}^{4}F^{-2}, \\
   H=bC_{1}^{3}t^{-1}EF^{-2},
\end{gathered} \label{3.4e3}
\end{equation}
where $a$ and $b$ are also arbitrary but satisfy
\begin{equation}\label{3.4e4}
    \frac{\beta}{2 \delta}=a^{-3/2}b^{2},
\end{equation}
and
\begin{equation}\label{3.4e}
    E=\sqrt{t(t+1)}-\ln(\sqrt{t}+\sqrt{t+1})+C_{2}, \quad
    F=E\sqrt{\frac{t+1}{t}}-t.
\end{equation}
Applying the inverse transformation of \eqref{ee9} to formula
\eqref{3.4e3}, namely
\[
    \tau=q^{-1/2}, \quad H=u\tau^{-3/2},
\]
we have
\begin{equation}
\begin{gathered}
  \tau=a^{-1/2}C_{1}^{-2}F, \\
   u=\tau^{\frac{3}{2}}bC_{1}^{3}t^{-1}EF^{-2}.
\end{gathered} \label{3.4e5}
\end{equation}
Further, applying the inverse transformation of \eqref{ec8} to
formula \eqref{3.4e5}, under the given parametric condition
\eqref{3.4e1}, we obtain the solution for equation \eqref{e23} in
the parametric form as follows:
\begin{gather*}
   \xi=\frac{\ln\big(aC_{1}^{4}F^{-2}\big)}{2\delta}, \\
   u=e^{-\frac{3}{2}\delta\xi}bC_{1}^{3}t^{-1}EF^{-2}\label{3.4e6},
\end{gather*}
where $a$ and $b$ are arbitrary constants, and satisfy condition
\eqref{3.4e4}.


\noindent {\bf Case 3:} Assume that
\begin{equation}\label{3.4e7}
    m=-3, \quad \kappa=-3, \quad \beta\delta=-1.
\end{equation}
In this case,  \eqref{e24} takes the form
\begin{equation}\label{3.4e8}
    \ddot{H}=AqH^{-3}\dot{H}-AH^{-2},
\end{equation}
where $A=-\frac{\beta}{3 \delta}$.

We know that the solution of equation \eqref{3.4e8} can be
expressed in the parametric form
\begin{equation} \label{3.4e9}
\begin{gathered}
  q=aC_{1}^{3}F^{-1}\sqrt{\frac{t+1}{t}}, \\
  H=bC_{1}^{2}F^{-1},
\end{gathered}
\end{equation}
where $F$ is the same as that in \eqref{3.4e}, $C_1$ and $C_2$ are
arbitrary constants, $a$ and $b$ are also arbitrary but satisfy
\begin{equation}\label{3.4e10}
    \frac{\beta}{3 \delta}=2a^{-2}b^{3}.
\end{equation}
Applying the inverse transformation of \eqref{ee9} to formula
\eqref{3.4e9}, namely
\[
    \tau=q^{-1/3}, \quad H=u\tau^{-2},
\]
we have
\begin{equation}
\begin{gathered}
   \tau=a^{-1/3}C_{1}^{-1}F^{1/3}
  \big(\frac{t+1}{t}\big)^{-1/6}, \\
   u=\tau^{2}bC_{1}^{2}F^{-1}.
\end{gathered} \label{3.4e11}
\end{equation}
Further, applying the inverse transformation of \eqref{ec8} to
formula \eqref{3.4e11}, under the given parametric condition
\eqref{3.4e7}, we obtain the solution for equation \eqref{e23} in
the parametric form as follows:
\begin{gather*}
 \xi=\frac{\ln\big(aC_{1}^{3}F^{-1}\sqrt{\frac{t+1}{t}}\big)}
{3\delta}, \\
  u=e^{-2\delta\xi}bC_{1}^{2}F^{-1}\label{3.4e12},
\end{gather*}
where $a$ and $b$ are arbitrary constants, and satisfy condition
\eqref{3.4e10}.



\section{Conclusion}


Finding first integrals (conservation laws) and exact solutions
for various nonlinear differential equations has been an
interesting subject in mathematical and physical communities.
Since 1983, Prelle and Singer presented a deductive method for
solving first--order ODEs that presents a solution in terms of
elementary functions if such a solution exists. This technique has
attracted many researchers from diverse groups and has been
extended to autonomous systems of ODEs of higher dimensions for
finding the first integrals and exact solutions under certain
assumptions.  From illustrative examples in these works, the
obtained first integrals of autonomous systems are usually of
rational or quasi-rational forms and searching for solution sets
$(S, R)$ usually involves complicated calculations. However, the
generalization of this procedure to autonomous/nonautonomous
systems of higher dimensions to find elementary first integrals in
an effective manner is still an interesting and important subject.


In this paper, we showed that under certain parametric conditions,
some new first integrals of the Duffing--van der Pol--type
oscillator equation \eqref{ee1} could be established. To reach our
goal, we first made a series of nonlinear transformations to
simplify equation \eqref{ee1} to a simple form, then by means of
the Preller--Singer method we derived the first integral of the
resultant equation. Through the inverse transformations we obtain
the first integrals of the original oscillator equations. Finally,
using the established first integral, we obtain exact solutions of
equation \eqref{ee1} in the parametric forms. Some statements in
the literature are corrected and clarified.

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\end{document}