International Journal of Analysis and Applications

Volume 19, Number 6 (2021), 970-983

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-970

ON TRULY NONLINEAR OSCILLATOR EQUATIONS OF ERMAKOV-PINNEY TYPE

MARCELLIN NONTI, KOLAWOLE KEGNIDE DAMIEN ADJAÏ, JEAN AKANDE,

MARC DELPHIN MONSIA∗

Department of Physics, University of Abomey-Calavi, Abomey-Calavi, 01.BP.526, Cotonou, Benin

∗Corresponding author: monsiadelphin@yahoo.fr

Abstract. In this paper we present a general class of differential equations of Ermakov-Pinney type which

may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration

after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the

first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial

differential equations.

1. Introduction

In the research field of periodic solution to Lienard nonlinear differential equations of the form

(1.1) ẍ + f(x) = 0

where the overdot stands for the derivative with respect to time, and f(x) is a nonlinear function of x,

it is less usual to notice differential equations with exact periodic solutions. It is again very less usual to

find differential equations with exact periodic solutions in terms of trigonometric functions. This makes the

Ermakov-Pinney equation

(1.2) ẍ + ax +
b

x3
= 0

Received October 5th, 2021; accepted November 2nd, 2021; published November 19th, 2021.

2010 Mathematics Subject Classification. 34C15, 34C25.

Key words and phrases. Lienard equations; Ermakov-Pinney equation; truly nonlinear oscillators; periodic solution.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

970

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-970


Int. J. Anal. Appl. 19 (6) (2021) 971

unusual and underlines its high usefulness in science and engineering. In this way a lot of applications in

classical mechanics as well as in quantum mechanics for example, has been carried out during these decades

( [1], [2]). The Ermakov-Pinney equation (1.2) has been studied in [2] to show the existence of new periodic

solutions and non-periodic solutions. In [3] an exceptional Lienard equation with strong and high order

nonlinearity is presented. A Lienard equation with a periodic solution in terms of a single trigonometric

function, which may lead to a quadratic Lienard type equation with a periodic solution exhibiting harmonic

oscillations, and contains several well-known equations like the Ermakov-Pinney equation [4], the Mickens

truly nonlinear oscillators and the cubic Duffing equation as special cases, has never been highlighted in the

literature despite the well established theory of differential equations, as it is carried out in [3]. According

to ( [5], [6]) all Ermakov-Pinney equations may be reduced to

(1.3) ẍ +
b

x3
= 0

using a variable change. One may say that the cubic singularity defines the nonlinear property of the

Ermakov-Pinney equation. So a differential equation with cubic nonlinearity may be said of Ermakov-Pinney

type. In this perspective consider the Lienard differential equation ( [2], [3])

(1.4) ẍ +
1

2
(α−q)axα−q−1 +

qb

2
x−q−1 = 0

Making q = 2, yields as equation

(1.5) ẍ +
1

2
(α− 2)axα−3 +

b

x3
= 0

In view of the above, the equation (1.5) may be characterized of Ermakov-Pinney type. For α = 2, the

equation (1.5) reduces to (1.3). The equation (1.5) may be reduced to a quadrature. Using the corresponding

first integral ( [2], [3])

(1.6) ẋ2x2 + axα = b

the general solution of (1.5) is written as the quadrature defined by

(1.7) ±(t + K) =
∫

xdx
√
b−axα

Let

(1.8) J =

∫
xdx

√
b−axα



Int. J. Anal. Appl. 19 (6) (2021) 972

The change of variable X2 = b − axα, and dx = − 2
aα

(
b−X2
a

)1−α
α

XdX, where a 6= 0, and α 6= 0, turns J

into

(1.9) J = −
2

αa
2
α

∫ (
b−X2

)2−α
α

dX

A new change of variable φ = X√
b
, where b 6= 0, reduces J to the form

(1.10) J = −
2
√
b

bα

(
b

a

) 2
α
∫ (

1 −φ2
)2−α

α

dφ

As can be seen, the value of the integral in J could not be known exactly. A new change of variable in

terms of trigonometric or hyperbolic functions may be also performed but this does not solve the problem.

However, it shows that the general solutions of some specific equations of the equation (1.5) are not periodic.

The equation (1.5) may be reduced to the form

(1.11) ẍ + (n− 1)ax2n−3 +
b

x3
= 0

where α = 2n, n is an integer. The equation (1.11) may be of physical importance since it has the structure

of truly nonlinear oscillators formulated by Mickens in his book [7], and contains the famous Ermakov-

Pinney equation as special case. It is known also that differential equations with power nonlinearities are

often encountered in mathematical modeling of physical problems. A vast literature exists on the topic of

truly nonlinear oscillators. During the last decades many authors investigated these nonlinear differential

equations. As nonlinear differential equations, they have no exact explicit solutions in general. Moreover

they could not be solved by the well known standard approximate analytical techniques [7]. So the existence

of periodic solutions of these equations is yet under some debate. This particularly, becomes an attractive

research problem when the second order autonomous truly nonlinear equation has a singularity at the origin

and can have no critical point, necessary condition, according to [8], for a planar autonomous systems to

have a periodic solution. It was the case of the so-called pseudo-oscillator investigated in [8]. The authors [8]

concluded that such a differential equation has no periodic solution. In contrast to this the author in [9]

showed that periodic solution exists, at least a non-smooth solution. The author [9] carried out a theory to

build such periodic solutions. In [10] the two general solutions predicted in [8] have been exactly calculated

and the authors [10] concluded also to the non existence of smoth periodic solution. The equation (1.11)

has a cubic singularity at the origin for a positive integer n, but may have fixed points. For n negative,

singularities appear also. Choosing n = 1, that is α = 2, reduces the equation (1.11) to the Ermakov-Pinney



Int. J. Anal. Appl. 19 (6) (2021) 973

type equation (1.3). For n = 2, that is α = 4, the equation (1.11) becomes the Ermakov-Pinney equation

(1.2). Using n = 3, in other words α = 6, the equation (1.11) transforms into

(1.12) ẍ + 2ax3 +
b

x3
= 0

For b = 0, the equation (1.12) reduces to the well-known restricted cubic Duffing equation ( [7], [11], [12])

for which it is said that all the solutions are periodic. Contrary to these authors and to several others, it has

been shown that such an equation may exhibit non-periodic solution, precisely complex-valued solutions [13].

Putting n = 4, into the equation (1.11), yields

(1.13) ẍ + 3ax5 +
b

x3
= 0

When b = 0, the equation (1.13) reduces to the restricted quintic Duffing equation. The equations (1.12),

(1.13) and others, which may be obtained for various values of n, have not been previously studied in

the literature. The above underlines the mathematical importance to ask whether the equation (1.11) has

periodic solutions for n � 2. The case n = 2, corresponding to the Ermakov-Pinney equation has been

investigated in several papers. Periodic solutions have been carried out for this equation in ( [2], [4]). The

case of negative n will be considered separately in a paper. In the present work we show that the equation

(1.11) may exhibit periodic and explicit general solutions for a
b

= 1, and 2 � n � 4. The choice of negative b

has no physical sense as we can see below. In this regard the qualitative properties of solutions of the equation

(1.11) are investigated in section (2), and the section (3) is devoted to exhibit exact periodic solutions of

(1.11) under the preceding conditions. The general solutions of the equation (1.3) are also exhibited in this

section. Finally a conclusion is formulated for the work.

2. Qualitative properties of solutions

The qualitative properties of solutions to (1.11) are investigated in this section using the phase plane

method. Therefore the equation (1.11) is equivalent to the planar autonomous dynamical system

(2.1)




ẋ = y

ẏ = −(n− 1)ax2n−3 − b
x3

The fixed point is defined by y = 0 and x =

[
− b

(n−1)a

] 1
2n

. As one may see, for b
a

= −1, the critical point

is real, but for b
a

= 1, the coordinate x may become complex. From (2.1) one may write



Int. J. Anal. Appl. 19 (6) (2021) 974

(2.2)
dy

dx
= −

(n− 1)ax2n + b
yx3

The separation of variable leads to

ydy = −
[
(n− 1)ax2n−3 + b

x3

]
dx

By integration, one may obtain the integral curves given by

(2.3) y2 = −ax2n−2 +
b

x2
+ c

This means, according to the equation (1.6) that the integration constant can be choosen as c = 0, so that

the Hamiltonian of the system can be written

(2.4) H = ẋ2x2 + ax2n

The Hamiltonian (2.4) is time independent such that the equation (1.11) defines a conservative system from

physical point of view. In this context, one can expect the existence of periodic solutions to the equation

(1.11). Now the objective is to calculate the exact and explicit general solutions of the equations (1.3), (1.12)

and (1.13).

3. General solutions

3.1. General solution of (1.3). The reduced Ermakov-Pinney equation (1.3) has been the object of a high

consideration in the literature since the Ermakov-Pinney equations can be reduced to (1.3) as mentioned

before. In order to study its properties, the Ermakov-Pinney equation (1.3) for b = −1, in ( [5], [6]) is

transformed into

(3.1) 2zz̈ − 3ż2 + 4z4 = 0

using the point transformation

(3.2) z =
1

x2

However, no explicit general solution is given. In [6] the equation (3.1) is examined from symmetry group

point of view and the authors [6] arrived to examine

(3.3) 2zz̈ − 3ż2 + Bz4 = 0



Int. J. Anal. Appl. 19 (6) (2021) 975

where B is the arbitrary constant. The authors [6] observe that the analysis of Lie point symmetries is

not adequate for (3.3) which requires, rather than nonlocal symmetry calculation. As the evaluation of

nonlocal symmetries may be complicated, the authors [6] apply the Jacobi last multiplier approach to find

the solution of (3.1) in terms of time dependent integral. On the other hand, the equation (1.3) is also

investigated in [14]. The authors [14] succeed to calculate a general solution of (1.3) where b = 1, in terms

of the so-called C invariant related to the Ermakov-Pinney invariant. However, the point transformation of

the equation (1.3) into (3.1) shows clearly that such an equation has two general solutions which differ only

by a sign. The objective in this paragraph is to determine in a direct fashion the two general solutions and

by exactly solving the auxiliary equation (3.1), in terms of arbitrary constants.

3.1.1. Direct method. According to (1.10) the equation (1.3) may be reduced to the quadrature

(3.4) ±
(
t + K

)
= −
√
b

b

(
b

a

)∫
dφ

which leads to

(3.5) −
√
b

b

(
b

a

)
φ = ±

(
t + K

)
Using the change φ = X√

b
= X

√
b

b
, the equation (3.5) gives

(3.6) X = ±a(t + K)

from which, using the previous relation X2 = b−ax2, one may secure the general solutions of (1.3) as

(3.7) x(t) = ±
[
b

a
−a(t + K)2

]1
2

where K is an integration constant.

3.1.2. Solution using the auxiliary equation (3.1). Let us consider the generalized Sundman transformation

theory introduced recently in the literature by Akande and coworkers [15]. In fact the generalized Sundman

transformation is a powerfull change of variables which allows solving differential equations with a few

mathematical manipulations. In the theory introduced by Akande et al. [15] the oscillator harmonic equation

(3.8) ü + ω2u = 0

where ω is a constant, ü means the second derivative with respect to τ, and

(3.9) u(τ) = A0sin(ωτ + β)

is transformed, under the change of variables



Int. J. Anal. Appl. 19 (6) (2021) 976

(3.10) u(τ) = F(t,z), dτ = G(t,z)dt

where

(3.11) F(t,z) =

∫
g(z)ldz, G(t,z) = eγϕ(z), and

∂F(t,z)

∂z
6= 0

to the second order differential equation

(3.12) z̈ +

(
l
g′(z)

g(z)
−γϕ′(z)

)
ż2 + ω2

exp(2γϕ(z))
∫
g(z)ldz

g(z)l
= 0

where A0, β, l, γ are arbitrary parameters, g(z) 6= 0, and ϕ(z) are arbitrary functions of z, and prime denotes

differentiation with respect to z. The application of ϕ(z) = ln(f(z)), leads to

(3.13) z̈ +

(
l
g′(z)

g(z)
−γ

f′(z)

f(z)

)
ż2 + ω2

f(z)2γ
∫
g(z)ldz

g(z)l
= 0

Putting g(z) = z, and f(z) = z2, into (3.13), allows one to obtain

(3.14) z̈ + (l− 2γ)
ż2

z
+

ω2

l + 1
z4γ+1 = 0

which may reduce, choosing γ = −l = 1
2
, to

(3.15) z̈ −
3

2

ż2

z
+ 2ω2z3 = 0

The equation (3.15) may be identified to (3.3) when 4ω2 = B and to (3.1) when ω2 = 1. So from the

solutions of (3.15) one may deduce those of (1.3) with b = −1. In this context the transformation defined

by (3.10) and (3.11) becomes

u(τ) =
∫
z−

1
2 dz = 2z

1
2

that is, z = u
2

4
, and dτ = zdt , so that the equation (3.15) may be reduced to the quadrature determined

by

(3.16)
A20
4

(t + K1) =

∫
dτ

sin2(ωτ + β)

where K1 is an integration constant. The change of variable s = ωτ + β, reduces the integral in (3.16) to



Int. J. Anal. Appl. 19 (6) (2021) 977

(3.17) J =
1

ω

∫
ds

sin2(s)

which yields

(3.18) −cot(s) =
ωA20

4
(t + K1)

such that

(3.19) ωτ + β = cot−1
[
−
ωA20

4
(t + K1)

]
Substituting (3.19) into (3.9) yields the general solution to (3.15) as

(3.20) z(t) =
1

4
A20sin

2

[
cot−1

(
−
ωA20

4
(t + K1)

)]
Using (3.20) one may deduce the solution of (3.1) in the form

(3.21) z(t) =
1

4
A20sin

2

[
cot−1

(
−
A20
4

(t + K1)
)]

Therefore the solution of (1.3) where b = −1, becomes

(3.22) x(t)2 =
4

A20sin
2

[
cot−1

(
− A

2
0

4
(t + K1)

)]
that is

(3.23) x(t) = ±
2

A0sin

[
cot−1

(
− A

2
0

4
(t + K1)

)]
Knowing −cot−1(ν) = sin−1

(
1√

1+ν2

)
, for ν � 0, and cot−1(ν) = π−sin−1

(
1√

1+ν2

)
, for ν ≺ 0, then the

general solutions (3.23) reduce to

(3.24) x(t) = ±
2

√
1 +

A40
16

(t + K1)2

A0

Now the problem to be solved is to calculate the general solution of the equations (1.12) and (1.13) to show

analytically and explicitly the existence of periodic solutions to the equation (1.11).



Int. J. Anal. Appl. 19 (6) (2021) 978

3.2. Exact periodic and complex-valued solutions.

3.2.1. Periodic and complex-valued solutions of (1.12). The equation (1.12) is obtained when n = 3, that is

when α = 6, from the equation (1.11). Two cases may be investigated.

Periodic solution

For reason of simplicity we choose a = b = 1. In this case the integral J becomes

(3.25) J = −
1

3

∫
dφ

3
√

(1 −φ2)2

which may be rewritten as [17]

(3.26) J = −
1

3

∫ φ
0

dξ
3
√

(1 − ξ2)2
= ±(t + c1)

where 0 ≺ φ ≺∞, and c1 is an arbitrary parameter.

By integration, (3.26) reduces to [17]

(3.27) J =
1

3

3

2
4
√

3
F(ψ,k)

where ψ = cos−1
[√

3−1− 3
√
φ2−1

√
3+1+ 3

√
φ2−1

]
, and k2 = 2+

√
3

4
. Using (3.27), one may write

(3.28) J =
1

2
4
√

3
F(ψ,k) = ±(t + c1)

from which one may get

cosψ = cn[±2 4
√

3(t + c1),k]

that is

(3.29)

√
3 − 1 − 3

√
φ2 − 1

√
3 + 1 + 3

√
φ2 − 1

= cn
[
± 2 4
√

3(t + c1),k
]

In this situation φ may be written as

(3.30) φ(t) = ±


1 +

(√
3 − 1 − (

√
3 + 1)cn

[
± 2 4
√

3(t + c1),k
]

1 + cn
[
± 2 4
√

3(t + c1),k
]

)3
1
2

Knowing φ = X√
b

= X, that is xα = b−X
2

a
, which becomes xα = 1 −X2, as a = b = 1, the solution x takes

the definitive form

(3.31) x(t) =

(
(1 −

√
3) + (

√
3 + 1)cn

[
± 2 4
√

3(t + c1),k
]

1 + cn
[
± 2 4
√

3(t + c1),k
] )12



Int. J. Anal. Appl. 19 (6) (2021) 979

Complex-valued solution: a = −b = 1

This case corresponds to

(3.32) J =
i

3

∫
(1 −φ2)

−2
3 dφ =

i

3

∫
dφ

3
√

(1 −φ2)2

where i is the purely imaginary number. The equation (3.32) gives [17]

(3.33)

∫ φ
0

dξ
3
√

(1 − ξ2)2
= ±3i(t + c2)

where 0 ≺ φ ≺∞, and c2 is an arbitrary parameter.

The evaluation of the integral in (3.33) leads to

(3.34)
3

2
4
√

3
F(ψ,k) = ±3i(t + c2)

The equation (3.34) may be rewritten in the form

(3.35) cosψ = cn

[
± 2i 4
√

3(t + c2),k

]
such that

(3.36)

√
3 − 1 − 3

√
φ2 − 1

√
3 + 1 + 3

√
φ2 − 1

= cn

[
± 2i 4
√

3(t + c2),k

]
From (3.36) one may get φ as

(3.37) φ(t) = ±
[
1 +

(
(
√

3 − 1)cn
[
± 2 4
√

3(t + c2),k
′
]
− (
√

3 + 1)

1 + cn
[
± 2 4
√

3(t + c2),k′
]

)3]1
2

with k′ =
√

1 −k2.

In the present case, φ = X√−1 , which is rewritten as φ = −iX, that is X = iφ, so that

(3.38) xα = −1 −X2 = φ2 − 1

from which one may secure the complex-valued solution x(t) in the form

(3.39) x(t) =

[
(
√

3 − 1)cn
[
± 2 4
√

3(t + c2),k
′
]
− (
√

3 + 1)

1 + cn
[
± 2 4
√

3(t + c2),k′
] ]12



Int. J. Anal. Appl. 19 (6) (2021) 980

3.2.2. Periodic and complex-valued solutions of (1.13). The equation (1.13) corresponds to n = 4, which

gives α = 8. Two cases may be studied.

Periodic solution: a = b = 1

In this case, the integral J is written in this form

(3.40) J = −
1

4

∫
dφ

4
√

(1 −φ2)3

Using (1.7) one may get the equation ( [16], [17])

(3.41) J = −
1

4

∫ φ
0

dξ
4
√

(1 − ξ2)3
= ±(t + c3)

where c3 is an arbitrary parameter. Therefore, by integration, the equation (3.41) reduces to

(3.42)
√

2F(ψ,

√
2

2
) = ∓4(t + c3)

where 0 ≺ φ � 1. From (3.42) one may ensure the following equation

(3.43) cosψ = cn

[
∓ 2
√

2(t + c3),

√
2

2

]
which is written in this form

(3.44)
4
√

1 −φ2 = cn
[
∓ 2
√

2(t + c3),

√
2

2

]
From the equation (3.44), one may secure

(3.45) φ2 = 1 − cn4
[
2
√

2(t + c3),

√
2

2

]
Using the relation φ = X√

b
, that is X = φ, the equation (3.45) is rewritten as

(3.46) X2 = 1 − cn4
[
2
√

2(t + c3),

√
2

2

]
from which the relation xα = b−X

2

a
, that is x8 = 1 −X2, gives

(3.47) x(t) =

[
cn

(
2
√

2(t + c3),

√
2

2

)]12
Complex-valued solution: a = −b = 1

This condition leads to the equation [17]



Int. J. Anal. Appl. 19 (6) (2021) 981

(3.48)

∫ x
0

ξdξ√
1 + ξ8

= ±i(t + c4)

where 0 ≺ x � 1. The integral in (3.48) is known as hyperelliptic integral and its evaluation gives [18]

(3.49)
1

4
cn−1

[
1 −x4

1 + x4
,

√
2

2

]
= ±i(t + c4)

which may be rearranged in the form

(3.50)
1 −x4

1 + x4
= cn

[
±4i(t + c4),

√
2

2

]
From (3.50) one may get

(3.51) x4 =
1 − cn

[
4i(t + c4),

√
2
2

]
1 + cn

[
4i(t + c4),

√
2
2

]
so that the general solution x(t) may take the expression

(3.52) x(t) =


1 − cn

[
4i(t + c4),

√
2
2

]
1 + cn

[
4i(t + c4),

√
2
2

]



1
4

or in the definitive form

(3.53) x(t) =


cn

[
4(t + c4),

√
2
2

]
− 1

cn
[
4(t + c4),

√
2
2

]
+ 1




1
4

Thus, the above shows that under the conditions that a
b

= 1, and 2 � n � 4, the explicit general solutions

of (1.11) are periodic.

In the sequel of this work, the related quadratic Lienard type equations to the equation (1.13) is examined.

4. Quadratic Lienard type equations

To determine the quadratic Lienard type equations related to (1.13), consider the change of variable

(4.1) ϑ = xp

Thus one may obtain

(4.2)
dx

dt
=

1

p
ϑ̇ϑ

1−p
p



Int. J. Anal. Appl. 19 (6) (2021) 982

and

(4.3)
d2x

dt2
=

1

p
ϑ̈ϑ

1−p
p +

1 −p
p2

ϑ̇2ϑ
1−2p
p

Substituting (4.3) into (1.13) and taking into account (4.1), yields

(4.4) ϑ̈ +
1 −p
p

ϑ̇2

ϑ
+ 3apϑ

p+4
p + bpϑ

p−4
p = 0

The solution (3.47) ensures the general periodic solution of (4.4) in the form

(4.5) ϑ(t) =

(
cn

[
2
√

2(t + c3),

√
2

2

])p
2

where a = b = 1.

By application of p = 2, the equation (4.4) reduces to

(4.6) ϑ̈−
1

2

ϑ̇2

ϑ
+ 6ϑ3 +

2

ϑ
= 0

and its general solution becomes

(4.7) ϑ(t) = cn

[
2
√

2(t + c3),

√
2

2

]

which shows for the first time that the Jacobi elliptic function cn [18] may be solution of second order

autonomous non-polynomial differential equations. Now a conclusion of this work may be addressed.

5. Conclusion

In this paper a general class of truly nonlinear oscillator equations is presented. The conditions of existence

of periodic solutions are shown and periodic and explicit general solutions are examined. The general

solutions of a well known Ermakov-Pinney type equation are also calculated. Finally it has been shown

that the Jacobi elliptic function cn may be solution of second-order autonomous non-polynomial differential

equations.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.



Int. J. Anal. Appl. 19 (6) (2021) 983

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Publishing, 2015.

http://arxiv.org/abs/1612.07080
https://vixra.org/pdf/2010.0195v1.pdf
https://vixra.org/pdf/2010.0195v1.pdf
https://vixra.org/pdf/2011.0050v3.pdf
https://doi.org/10.16943/ptinsa/2020/154987
https://doi.org/10.16943/ptinsa/2020/154987
https://vixra.org/pdf/2009.0174v1.pdf
https://vixra.org/pdf/2009.0174v1.pdf

	1. Introduction
	2. Qualitative properties of solutions
	3. General solutions
	3.1. General solution of (1.3)
	3.2. Exact periodic and complex-valued solutions

	4. Quadratic Lienard type equations
	5. Conclusion
	References