# Parametric factorization of non linear second order differential equations

Gabriel González  Cátedra CONAHCYT–Universidad Autónoma de San Luis Potosí,
San Luis Potosí, 78000 MEXICO  Coordinación para la Innovación y la Aplicación
de la Ciencia y la Tecnología, Universidad Autónoma de San Luis Potosí,San
Luis Potosí, 78000 MEXICO

###### Abstract

In this paper the factorization method introduced by Rosu & Cornejo-Pérez for
second order non linear differential equations is generalized by adding a
parameter in order to obtain the general solutions for the mixed quadratic and
linear Liénard type equation. The new parametric factorization is used to
obtain complete analytic solutions for nonlinear second order differential
equations. The parametric factorization introduced in this article reduces to
the standard factorization scheme when the parameter goes to zero. As an
example, we apply the parametric factorization approach to solve the
generalized Fisher equation and the Israel-Stewart cosmological model. The
parametric factorization presented in this paper can be used in other non
linear mixed Liénard type equations.

##  I Introduction

Non linear second order differential equations are widely used to describe
various phenomena in physics and mathematics and the vast majority of them do
not have analytic solutions and are very difficult to analyze them. Developing
methods for finding solutions for non linear differential equations has been a
problem of interest for a long time. At the present time there are many
methods for finding exact solutions of non linear equations such as: the
$\tanh$-expansion method [1], the Darboux transformation [2], the Bäcklund
transformation [3], Hirota bilinear method [4], Painlevé truncation expansion
[5, 6], generalized Sundman transformation [7], point transformations and
contact transformations[8]. All these approaches have yielded many interesting
exact solutions of the kink and soliton type for well-known nonlinear
equations.
Despite the vast and rich variety of different methods to solve non linear
differential equations, the fundamental problem of finding explicit and exact
analytic solutions to nonlinear differential equations constitutes yet an
active area of research. In the process of learning to solve non linear
differential equations it is convenient to begin with simple and efficient
methods which provide a way to obtain exact analytic solutions. One of the
most popular existing simple methods to solve non linear differential
equations is by using travelling wave transformations and direct integration.
However, the travelling wave transformation method is only suitable for a
certain type of non linear differential equations. Recently, Rosu & Cornejo
proposed a factorization method which allows one to obtain travelling wave
solutions of the reaction-diffusion equations with polynomial
nonlinearities[9]. Using the factorization method, Cornejo & Rosu obtained
particular solutions of several important equations, among which are the
Fisher equation, the FitzHugh-Nagumo equation and the generalized Burgers-
Huxley equation, which maps into a second order non linear differential
equation of Liénard type in the travelling coordinate frame of the form[10]

$\ddot{x}+f(x)\dot{x}+g(x)=0$ (1)

where $f(x)$ and $g(x)$ are polynomial functions and overdot denotes
differentiation with respect to time.
In this paper we extend the factorization method introduced by Rosu & Cornejo
to solve a mixed quadratic-linear Liénard type equation of the form

$\ddot{x}+\mu\frac{\dot{x}^{2}}{x}+F(x)\dot{x}+G(x)=0$ (2)

where $\mu$ is an auxiliary parameter to be determined. One can see that
equation (1) is a subcase of equation (2). In applications one often
encounters differential equations in which both linear and quadratic terms are
present. Equation (2) has a particular form of a mixed type Liénard equation
which frequently appears as a mathematical model in several areas of physics,
for example equation (2) belongs to the type of second order Gambier equation
when the coefficients are assumed to be constant parameters. The Gambier
equation written as a second order differential equation takes the form[11]

$\displaystyle\ddot{x}=$
$\displaystyle\frac{n-1}{n}\frac{\dot{x}^{2}}{x}+\left(\frac{n+2}{n}ax-\frac{n-2}{n}\frac{\sigma}{x}+b\right)\dot{x}$
(3)
$\displaystyle-\frac{a^{2}}{n}x^{3}+(\dot{a}-ab)x^{2}+\left(cn-\frac{2a\sigma}{n}\right)x-b\sigma-\frac{\sigma^{2}}{nx}.$
(4)

where $a$, $b$ and $c$ are functions of the independent variable and $\sigma$
is a constant. Interestingly, the parametric factorization introduced in this
paper is of the type of an autonomous second order Gambier equation. The
importance of the second order Gambier equation is due to the fact that it is
related with very important non linear differential equations, such as the
second order Ricatti equations, second order Kummer-Schwartz equation and
Milne-Pinney equation, to name a few[12]. More recently, Zheng and Shang[13]
showed that the amplitude part of the solution in phase amplitude format of
the nonlinear Schrödinger equation with dual power nonlinearities satisfies a
mixed Liénard type equation of the form given in equation (2). Thus, the goal
of this paper is to present an extension of the factorization method in which
one can obtain solutions of a mixed Liénard type equation. The parametric
factorization introduced in this paper has the advantage that allows to obtain
solutions of second order non linear differential equations with linear and
quadratic damping terms and contains as a particular case the standard
factorization.
The article is organized as follows. In the first section we will review the
Rosu & Cornejo-Pérez factorization scheme and introduce the parametric
factorization. In the second section we will apply the parametric
factorization to obtain particular and parametric solutions of the generalized
Fisher. In the third section we will use the parametric factorization to
obtain particular and parametric solutions for the Israel-Stewart cosmological
model. The conclusions are summarized in the last section.

##  II Parametric Factorization

An elegant procedure to solve second order non linear differential equations
consists in using the factorization method, where a given non linear
differential operator is factorized in two first order differential operators.
In 2005, Rosu and Cornejo-Pérez [9] introduced an effective factorization of
second-order ordinary differential equations with polynomial nonlinearities by
taking additional advantage from the polynomial factorization of the nonlinear
part. Using the factorization technique, Rosu and Cornejo-Pérez obtained
particular solutions of the following Liénard type differential equation

$\ddot{q}+f(q)\dot{q}+g(q)=0,$ (5)

where the dot represents the derivative with respect to time. Equation (5)
admits the following factorization

$\left(D-\phi_{2}(q)\right)\left(D-\phi_{1}(q)\right)q=0~{},\qquad
D=\frac{d}{dt}~{}.$ (6)

By expanding equation (6) and comparing with equation (5), one obtains the
following conditions over the functions $\phi_{1}$ and $\phi_{2}$:

$\displaystyle\phi_{1}+\phi_{2}+q\frac{d\phi_{1}}{dq}=-f(q)$ (7)
$\displaystyle\phi_{1}\phi_{2}=\frac{g(q)}{q}~{}.$ (8)

To obtain a particular solution, Rosu and Cornejo-Pérez solved the following
first order differential equation

$\left(D-\phi_{1}(q)\right)q=0$ (9)

obtaining a particular solution of (11) by one quadrature

$t-t_{0}=\int\frac{dq}{q\phi_{1}(q)}~{}$ (10)

In this article we will extend the factorization method outlined above by
adding a parameter $\mu$ in equation (6) such that the factorization is given
now by

$\left(D-\varphi_{2}(x)\right)\left(D-\varphi_{1}(x)\right)x^{\mu+1}=0,$ (11)

where $\mu\neq-1$. If we expand the factorization given in equation (11) we
get the following non linear second order differential equation

$\ddot{x}+\mu\frac{\dot{x}^{2}}{x}-\dot{x}\left(\varphi_{1}(x)+\varphi_{2}(x)+\frac{x}{\mu+1}\frac{d\varphi_{1}}{dx}\right)+\frac{x}{\mu+1}\varphi_{1}(x)\varphi_{2}(x)=0.$
(12)

Equation (12) is a particular form of the mixed Liénard type equation with
quadratic and linear terms. Note that equation (12) reduces to the standard
Liénard type equation when $\mu\rightarrow 0$. By comparing (2) and (12), one
obtains the conditions for the parametric factorization over the functions
$\varphi_{1}$ and $\varphi_{2}$:

$\displaystyle\varphi_{1}+\varphi_{2}+\frac{x}{\mu+1}\frac{d\varphi_{1}}{dx}=-F(x)$
(13) $\displaystyle\varphi_{1}\varphi_{2}=(\mu+1)\frac{G(x)}{x}~{}.$ (14)

An interesting feature between the standard factorization and the parametric
factorization is that we can transform one to the other by a non trivial space
transformation once written in factorized form. This result allows us to map
solutions into solutions between the Liénard equation (1) and the mixed
Liénard equation (2). Let us now work out a simple example to illustrate this
point. Consider the following Liénard type equation given by

$\ddot{q}+(2m+3)q^{2m+1}\dot{q}+q+q^{4m+3}=0,$ (15)

with $m$ a non negative integer. Equation (15) represents a class of solvable
nonlinear oscillators with isochronous orbits [14], i.e. orbits with fixed
period, not dependent with the amplitude. Equation (15) can be written in the
following standard factorization form

$\left(D+q^{2m+1}+i\right)\left(D+q^{2m+1}-i\right)q=0$ (16)

We can make the following space transformation $q^{2m+1}=x$ so that equation
(16) becomes

$\left(D+x+i\right)\left(D+x-i\right)x^{1/(2m+1)}=0$ (17)

Equation (17) is now written in the parametric factorization form where
$\mu=-2m/(2m+1)$. We can expand equation (17) to get

$\ddot{x}-\left(\frac{2m}{2m+1}\right)\frac{\dot{x}^{2}}{x}+(2m+3)x\dot{x}+(2m+1)(x+x^{3})=0.$
(18)

Equations (16) and (17) share the same solution given by

$q(t)=x^{1/(2m+1)}(t)=\frac{\sin(t-t_{0})}{\left(C+(2m+1)\int\sin^{2m+1}(t-t_{0})dt\right)^{1/(2m+1)}}$
(19)

where $t_{0}$ and $C$ are arbitrary constant. Therefore, the solution for the
mixed Liénard equation (18) is given by

$x(t)=\frac{\sin^{2m+1}(t-t_{0})}{C-\cos(t-t_{0})\sum_{r=0}^{m}A_{mr}\sin^{2r}(t-t_{0})}$
(20)

with

$A_{mr}=\frac{2^{2(m-r)}(m!)^{2}(2r)!}{(2m)!(r!)^{2}},$ (21)

where the condition for periodic solutions is $|C|>A_{m0}$ [14]. The solution
for the mixed Liénard type equation given in equation (18) are shown in figure
(1) for $m=1$ and $m=2$, respectively.

Figure 1: The figure shows the solutions $x(t)$ of the mixed Liénard type
equation for $m=1$ and $m=2$ respectively.

Consequently, solutions from the standard factorization scheme and their
properties can be used to obtain solutions of the parametric factorization
approach through a space coordinate transformation.

##  III Generalized Fisher equation

We will now show how to apply the parametric factorization approach to obtain
solutions of the generalized Fisher equation which is used in biology[15]. The
generalized Fisher equation is a non linear partial differential equation
which describes diffusion models for insects and biology invasion. From the
perspective of biology invasion, the generalized Fisher equation predicts how
the population of a particular species will spread via travelling waves. Let
us consider the generalized Fisher equation given by[16]

$\frac{\partial u}{\partial
t}=u^{p}\left(1-u^{q}\right)+\frac{\partial}{\partial
x}\left(u^{m}\frac{\partial u}{\partial x}\right)$ (22)

where $u$ represents the population density and $p$, $q$ and $m$ are positive
parameters. Solutions for equation (22) have been found for some values of the
parameters $p$, $q$ and $m$; in particular, the standard factorization method
has been used for the case $m=0$, $p=q=1$, i.e. the standard Fisher equation,
and for the case $m=0$, $p=1$ and $q=2$, i.e. the Burguers-Huxley equation,
respectively[10]. In this section we will consider the generalized Fisher
equation for the case $m\neq 0$ which can be written in the travelling
reference frame $\tau=kx-\omega t=k(x-vt)$ as the following mixed Liénard non
linear differential equation

$\ddot{u}+m\frac{\dot{u}^{2}}{u}+\omega\frac{\dot{u}}{k^{2}u^{m}}+\frac{u^{p-m}}{k^{2}}-\frac{u^{p+q-m}}{k^{2}}=0$
(23)

where the over-dot represents $D=\frac{d}{d\tau}$. Using the second
factorization condition given in equation (14) we have

$\varphi_{1}\varphi_{2}=\frac{1+m}{k^{2}}u^{p-(1+m)}\left(1-u^{q}\right).$
(24)

Therefore, we can choose

$\displaystyle\varphi_{1}$ $\displaystyle=$ $\displaystyle
a_{1}\frac{\sqrt{1+m}}{k}u^{(p-(1+m))/2}\left(1-u^{q/2}\right)$
$\displaystyle\varphi_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1+m}}{a_{1}k}u^{(p-(1+m))/2}\left(1+u^{q/2}\right)$

where $a_{1}$ is a nonzero constant to be determined with the first
factorization condition given in equation (13) which reads

$\displaystyle\varphi_{1}+\varphi_{2}+\frac{u}{\mu+1}\frac{d\varphi_{1}}{du}$
$\displaystyle=\frac{a_{1}}{k}u^{(p-(1+m))/2}\left[\sqrt{1+m}+\frac{p-(1+m)}{2\sqrt{1+m}}-u^{q/2}\left(\sqrt{1+m}+\frac{q-p+(1+m)}{2\sqrt{1+m}}\right)\right]$
(25)
$\displaystyle+\frac{\sqrt{1+m}}{a_{1}k}u^{(p-(1+m))/2}\left[1+u^{q/2}\right]=-\frac{\omega}{k^{2}u^{m}}$

It follows that we have to choose $p=1-m$ in order to satisfy the parametric
factorization conditions such that we get the following values for $a_{1}$ and
$\omega$

$a_{1}=\pm\sqrt{\frac{2(1+m)}{2+q}}\quad\mbox{and}\quad\omega=\mp\frac{k(q+4)}{\sqrt{2(q+2)}}.$
(26)

Therefore, the travelling wave solutions we are going to obtain are moving
with a constant velocity of $v=\omega/k=\mp\frac{(q+4)}{\sqrt{2(q+2)}}$, which
means that with increasing value of $q$ the velocity modulus increases from
$2$ to $\infty$.
Equation (23) admits the following parametric factorization

$\left[D\mp\sqrt{\frac{2+q}{2}}\frac{1}{ku^{2m}}\left(1+u^{q/2}\right)\right]\left[D\mp(1+m)\sqrt{\frac{2}{2+q}}\frac{1}{ku^{2m}}\left(1-u^{q/2}\right)\right]u^{m+1}=0.$
(27)

If one wants to find a particular solution to equation (23) we have to solve
only a compatible first order differential equation given by

$\left[D\mp(1+m)\sqrt{\frac{2}{2+q}}\frac{1}{ku^{2m}}\left(1-u^{q/2}\right)\right]u^{m+1}=0,$
(28)

which has the following implicit solution

$u^{2m}{{}_{2}}F_{1}\left[1,\frac{4m}{q},1+\frac{4m}{q},u^{q/2}\right]=\pm
2m\sqrt{\frac{2}{2+q}}(\tau-\tau_{0})$ (29)

where ${}_{2}F_{1}$ is the hypergeometric function and $\tau_{0}$ is an
integration constant. In figure (2) we show a travelling wave solution for the
generalized Fisher equation obtained with the parametric factorization
approach.

Figure 2: The figure shows the travelling wave solution $u(\tau)$ of the
generalized Fisher equation for $m=1/4$, $p=3/4$ and $q=2$, moving with
constant velocity of $v=3/\sqrt{2}$.

It is possible to solve the generalized Fisher equation given by

$\frac{\partial u}{\partial
t}=u^{1-m}\left(1-u^{q}\right)+\frac{\partial}{\partial
x}\left(u^{m}\frac{\partial u}{\partial x}\right)$ (30)

in a different way by using the parametric factorization conditions given in
equations (13) and (14) [17]. Solving $\varphi_{2}$ from the first equation
and substituting in the second equation we have

$-F(u)\varphi_{1}-\varphi_{1}\left(\varphi_{1}+\frac{u}{\mu+1}\frac{d\varphi_{1}}{du}\right)=(\mu+1)\frac{G(u)}{u},$
(31)

which is transformed into an Abel equation of the second kind

$ww^{\prime}=F(u)u^{\mu}w-G(u)u^{2\mu}$ (32)

by using the substitution

$w(u)=-\frac{u^{\mu+1}\varphi_{1}(u)}{\mu+1}.$ (33)

The Abel equation given in (32) admits exact parametric solutions for special
cases. For our particular case, we have $\mu=m$, $F(u)=\omega u^{-m}/k^{2}$
and $G(u)=u^{p-m}(1-u^{q})/k^{2}$, therefore we need to solve the following
Abel equation given by

$ww^{\prime}-\frac{\omega}{k^{2}}w=-\frac{u^{p-m}}{k^{2}}+\frac{u^{p-m+q}}{k^{2}}$
(34)

which for the case when $k^{-2}=2(2+q)/(4+q)^{2}$ and
$\omega=(4+q)^{2}/(4+2q)$ has the following solution in parametric form[18]

$u=\frac{(q+4)}{q}a\xi E^{2/q}_{1+q},\quad\mbox{and}\quad
w=aE^{2/q}_{1+q}\left(R_{1+q}E_{1+q}+\frac{2}{q}\xi\right)$ (35)

where

$E_{1+q}=\int(1\pm\xi^{2+q})^{-1/2}d\xi+C,\quad
R_{1+q}=\sqrt{1\pm\xi^{2+q}}\quad\mbox{and}\quad
a=\frac{4+q}{q}\left(\frac{2}{q}\right)^{2/q},$ (36)

where $C$ is an integral constant. Therefore, we have two different methods to
obtain the solution of the non linear differential equation, one gives a
particular solution and the other one gives a parametric solution.

##  IV Israel-Stewart Cosmological Model

A description of the relativistic thermodynamics of non-perfect fluids is
given by the so called Israel-Stewart cosmological model. For the case when
the bulk viscosity coefficient $\xi$ is given as a power law function of the
energy density by $\xi=\xi_{0}\rho^{1/2}$ a cosmological solution of the
polynomial type given by $H\propto(t+const.)^{-1}$ has been found by applying
the standard factorization method [19, 20], where $H$ denotes the Hubble rate
function. The nonlinear differential equation for the Hubble function is given
as the following mixed Liénard equation[21]

$\ddot{H}+\alpha_{1}\frac{\dot{H}^{2}}{H}+\alpha_{2}H\dot{H}+\alpha_{3}H^{3}=0$
(37)

where

$\displaystyle\alpha_{1}=$ $\displaystyle-\frac{3}{2\delta}$ (38)
$\displaystyle\alpha_{2}=$
$\displaystyle\frac{3}{2}+3(1+\omega)-\frac{9}{4\delta}(1+\omega)+\frac{\sqrt{3}\epsilon(1-\omega^{2})}{\xi_{0}}$
(39) $\displaystyle\alpha_{3}=$
$\displaystyle\frac{9}{4}(1+\omega)+\frac{9}{2}\epsilon(1-\omega^{2})\left[\frac{1+\omega}{\sqrt{3}\xi_{0}}-1\right]$
(40) $\displaystyle\delta(\omega)\equiv$
$\displaystyle\frac{3}{4}\left(\frac{1+\omega}{1/2+\omega}\right),$ (41)

are constant coefficients and $0\leq\omega<1$. In Ref.[19], Cruz et al.
factorized equation (37) in the following form

$\left(D-\phi_{1}(H)\dot{H}-\phi_{2}(H)\right)\left(D-\phi_{3}(H)\right)H=0,$
(42)

where, after some algebra they found the following factorization

$\left(D+\alpha_{1}\frac{\dot{H}}{H}-a^{-1}_{1}H\right)\left(D-a_{1}\alpha_{3}H\right)H=0,$
(43)

where

$a_{1}=\frac{-\alpha_{2}\pm\sqrt{\alpha^{2}_{2}-4\alpha_{3}(2+\alpha_{1})}}{2\alpha_{3}(2+\alpha_{1})}.$
(44)

Note that the factorization given in equation (42) uses three functions and
therefore it is more difficult to apply than the parametric factorization
introduced in this paper. A particular solution is obtained after solving the
first order differential equation given by

$\dot{H}-a_{1}\alpha_{3}H^{2}=0.$ (45)

In this section we are going to obtain the Hubble function by means of the
parametric factorization. By assuming that the functions have the following
form $\varphi_{1}=\tilde{a}_{1}\sqrt{(1+\alpha_{1})\alpha_{3}}H$ and
$\varphi_{2}=\tilde{a}^{-1}_{1}\sqrt{(1+\alpha_{1})\alpha_{3}}H$ and using the
parametric factorization conditions given in equations (13) and (14) we obtain

$\displaystyle\varphi_{1}(H)=$
$\displaystyle(1+\alpha_{1})\left[\frac{-\alpha_{2}\pm\sqrt{\alpha^{2}_{2}-4\alpha_{3}(2+\alpha_{1})}}{2(2+\alpha_{1})}\right]H$
(46) $\displaystyle\varphi_{2}(H)=$
$\displaystyle\frac{2\alpha_{3}(2+\alpha_{1})}{-\alpha_{2}\pm\sqrt{\alpha^{2}_{2}-4\alpha_{3}(2+\alpha_{1})}}H.$
(47)

Therefore, equation (37) admits the following parametric factorization

$\left(D-\frac{2\alpha_{3}(2+\alpha_{1})}{-\alpha_{2}\pm\sqrt{\alpha^{2}_{2}-4\alpha_{3}(2+\alpha_{1})}}H\right)\left(D-(1+\alpha_{1})\left(\frac{-\alpha_{2}\pm\sqrt{\alpha^{2}_{2}-4\alpha_{3}(2+\alpha_{1})}}{2(2+\alpha_{1})}\right)H\right)H^{1+\alpha_{1}}=0.$
(48)

To obtain a particular solution we have to solve the following first order
differential equation

$\left(D-(1+\alpha_{1})\left(\frac{-\alpha_{2}\pm\sqrt{\alpha^{2}_{2}-4\alpha_{3}(2+\alpha_{1})}}{2(2+\alpha_{1})}\right)H\right)H^{1+\alpha_{1}}=H^{\alpha_{1}}(1+\alpha_{1})\left[\dot{H}-a_{1}\alpha_{3}H^{2}\right]=0,$
(49)

which has the same solution as the previous standard factorization given by

$H(t)=\frac{A_{\pm}}{t-\left(t_{0}-\frac{A_{\pm}}{H_{0}}\right)},$ (50)

where $A_{\pm}=-(a_{1}\alpha_{3})^{-1}$ and $H_{0}=H(t_{0})$ is the Hubble
constant.
We are now going to solve equation (37) by changing it to an Abel equation as
we did in the previous section. By using equation (33) we arrive at the
following Abel equation of the second kind

$w\frac{dw}{dH}=\alpha_{2}H^{1+\alpha_{1}}w-\alpha_{3}H^{3+2\alpha_{1}}=\alpha_{2}H^{1+\alpha_{1}}\left(w-\frac{\alpha_{3}}{\alpha_{2}}H^{2+\alpha_{1}}\right).$
(51)

By making the transformation
$\eta=(\alpha_{2}/(2+\alpha_{1}))H^{2+\alpha_{1}}$, equation (51) becomes the
Abel equation in canonical form

$w\frac{dw}{d\eta}=w-\alpha_{3}\left(\frac{2+\alpha_{1}}{\alpha^{2}_{2}}\right)\eta,$
(52)

which has a solution in parametric form given by[18]

$\eta(\tau)=\frac{\alpha_{2}}{2+\alpha_{1}}H^{2+\alpha_{1}}(\tau)=C\exp\left(-\int\frac{\tau}{\tau^{2}-\tau-A}d\tau\right),\quad
w(\tau)=C\tau\exp\left(-\int\frac{\tau}{\tau^{2}-\tau-A}d\tau\right),$ (53)

where $C$ is a constant and $A=\alpha_{3}(2+\alpha_{1})/\alpha^{2}_{2}$.
It is interesting to note that the particular and parametric solutions satisfy
the following dynamic equation

$\frac{dH^{1+\alpha_{1}}}{dt}=-\left(1+\alpha_{1}\right)w\left(H(t)\right)$
(54)

where

$w(H)=-\frac{H^{1+\alpha_{1}}}{1+\alpha_{1}}\varphi_{1}(H)$ (55)

is the transformation used to obtain Abel´s equation. Therefore, using
equation (54) and comparing with equation (48) we can express the particular
solution as the following dynamic equation

$\frac{dH^{1+\alpha_{1}}}{dt}=-\left(1+\alpha_{1}\right)\left(1\pm\sqrt{1-\frac{4\alpha_{3}(2+\alpha_{1})}{\alpha^{2}_{2}}}\right)\eta.$
(56)

In figure (3) we show the graph $w$ vs. $\eta$ for the particular solution for
several values of $0\leq\omega<1$.

Figure 3: The figure shows the particular solution for the Hubble rate
function given in terms of $w$ as a function of $\eta$ for different values of
$0\leq\omega<1$. We have taken $\epsilon=\xi_{0}=3/4$.

In figure (4) we compare the particular and the parametric solutions where we
have taken $\alpha_{3}(2+\alpha_{1})/\alpha^{2}_{2}=1/5$. Note that the
particular solutions are the upper and lower curves of the figure.

Figure 4: The figure shows the particular (blue and orange lines) and
parametric solutions (green lines) given as a function of $\eta$ for the case
when $\alpha_{3}(2+\alpha_{1})/\alpha^{2}_{2}=1/5$ and $\epsilon=\xi_{0}=3/4$.
For the parametric solution we have used different values of the integration
constant $C$. The upper and lower lines correspond to the particular solution.

## Conclusions

In summary, compared with the factorization proposed by Rosu & Cornejo-Pérez
the parametric factorization presented in this paper is more general and
contains as a special case the standard factorization. The parametric
factorization can deal with a larger class of non linear second order
differential equations and also one can use the factorization conditions in
order to obtain parametric solutions for a given non linear differential
equation. Using the parametric factorization technique we have obtained a
particular solution and a parametric solution of the generalized Fisher
equation and the Israel-Stewart cosmological model in order to illustrate this
approach. The parametric factorization presented in this paper can be used in
other non linear mixed Liénard type equations.

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