LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción, Paraguay.

ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1640.

DOI: https://doi.org/10.56712/latam.v5i6.3112

Physical conditions in the analytical solution on the Mathieu
equation

Condiciones físicas en la solución analítica de la ecuación de Mathieu

Alberto Gutiérrez Borda
egutierrez@unica.edu.pe

https://orcid.org/0000-0001-6260-2419
Universidad Nacional San Luis Gonzaga

Ica – Perú

Orlando Berrocal Navarro
oberrocal@unica.edu

https://orcid.org/0000-0002-6151-6540
Universidad Nacional San Luis Gonzaga

Ica – Perú

Ricardo Cavero Donayre
ricardo.cavero@unica.edu

https://orcid.org/0000-0001-9020-870X
Universidad Nacional San Luis Gonzaga

Ica – Perú

Artículo recibido: 23 de noviembre de 2024. Aceptado para publicación: 07 de diciembre de 2024.
Conflictos de Interés: Ninguno que declarar.

Abstract

The study aims to analyze the Mathieu equation associated with linear systems under properties, use
the one-dimensional Floquet theorem and explain the behavior of the solutions of the Mathieu equation
and function relating to the Hill equation together with the theory of disturbances. The most common
mathematical problem in physics is finding solutions to certain second-order differential equations
subject to boundary conditions. A hypothetical deductive approach method, handling qualitative
equations of mathematical physics, was adopted in review. With Mathieu's equation, the existence of
the physical conditions that influence the value of the parameters that define the equation were
answered. It is important due to its usefulness in the field of analytical and mechanical dynamics,
physical systems subjected to parametric excitation; In addition, it leaves other variations open for
new applications.

Keywords: boundary problems, mathieu equation, hill equation, perturbation theory

Resumen
El estudio tiene como objetivo analizar la ecuación de Mathieu asociado a sistemas lineales bajo
propiedades, utilizar el teorema Floquet de una dimensión y explicar el comportamiento de las
soluciones de la ecuación y función de Mathieu relacionado con la ecuación de Hill junto a la teoría de
perturbaciones. Los problemas matemáticos más comunes en física son encontrar soluciones de
ciertas ecuaciones diferenciales de segundo orden sujetas a condiciones de frontera. Se adoptó en
revisión, un método enfoque hipotético deductivo, manejo de ecuaciones cualitativas de la física
matemática. Con la ecuación de Mathieu, se dio respuesta a la existencia de las condiciones físicas
que influyen para obtener el valor de los parámetros que definen la ecuación. Tiene importancia por
su utilidad en el ámbito de la dinámica analítica y mecánica, sistema físico sometidas a excitación

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ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1641.

paramétrica; además, deja abierta otras variaciones para nuevas aplicaciones.

Palabras clave: problemas de contorno, ecuación de mathieu, ecuación de hill, teoría de
perturbaciones

Todo el contenido de LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades,
publicado en este sitio está disponibles bajo Licencia Creative Commons.

Cómo citar: Gutiérrez Borda, A., Berrocal Navarro, O., & Cavero Donayre, R. (2024). Physical conditions
in the analytical solution on the Mathieu equation. LATAM Revista Latinoamericana de Ciencias
Sociales y Humanidades 5 (6), 1648 – 1654. https://doi.org/10.56712/latam.v5i6.3112

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ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1642.

INTRODUCTION

Most of the functions of physics-mathematics have their origin in research problems. Mathieu
functions were introduced when determining the modes of vibration of a stretching membrane having

an elliptical boundary. The two-dimensional wave equation was transformed into
��

2
��

��2 +
��2��
��2 + ��2�� = 0

confocal elliptic coordinates, and separated into two ordinary differential equations of the form:

��2��
��2 + (�� − 2��(2��))�� = 0 (1)

��2��
��2 − (�� − 2��(2��))�� = 0 (2)

being ��, �� real parameters.

Equation (2) was solved with substitution of ±�� by �� and vice versa. For �� > 0 (1) is the Mathieu
equation and (2) is the modified Mathieu equation; However, in the elliptical membrane problem, the
appropriate solutions to (2) are called Mathieu , periodic in �� with period �� or 2��, as a consequence of
this periodicity, �� has special values called characteristic numbers. The solutions of equation (2) that
correspond to the same solutions of (1) for the value of , are called modified �� Mathieu functions that
are obtained from the functions using imaginary arguments.

The first solutions of integral order of equation (1) were given through series of cosines and sines,
which satisfy the conditions to be Fourier series, although the coefficients are not obtained by ordinary
integration (Hille, 1923).

Mathieu's equation, that is
��2��
��2 + [�� − 2��(2��)]�� = 0, where −2��(2��) = 2[��2��2�� + ��4��3�� + ⋯ ], �� =

��0, being �� a known parameter.

In 1883, Floquet published a general work on linear differential equations with periodic coefficients, of
which Hill's and Mathieu 's equations are particular cases. The first appearance of an asymptotic
formula for modified Mathieu functions was given by Maclaurin; However, none of these authors
obtained the multiplicative constants, which are necessary for numerical work (Hartman, 1964;
Shingareva, 1995).

Marshall (1909) obtained the multipliers for the series, while Hilbert studied the characteristic values
and obtained an integral equation with discontinuous nuclei for the periodic solutions of equation (1).
In this same line, Sieger published an article on the diffraction of electromagnetic waves by an elliptical
cylinder and worked with orthogonality, using an integral equation with a different nucleus, and obtained
a solution of equation (2) as a series of products of Bessel functions, and the analysis of their
convergence.

A systematic study was given by Whittaker (1914), and he found an integral equation for a set of
periodic functions of integral order; Using this method as a basis, Young (1914) provided a method for
finding general solutions and discussed the problem of stability; That is, if the solution tends to zero or
infinity when �� → ∞, the recurrence formulas for the Mathieu functions were arrived at (Arnold, 1973).

A general study of the Mathieu equation is due to J. Dougall who obtained asymptotic expansions for
the Mathieu functions modified with ��large, a boundary integral, which under certain conditions,
degenerated to an integral of the Bessel function. Until 1921, the only known periodic solutions of the
Mathieu equation (1) had period �� y 2��. Poole recently generalized the situation and showed that with

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appropriate values of �� for an �� assigned, equation (1) can admit solutions that have period 2��, where
�� is an integer greater than or equal to two (Zimmerman, 1995).

The second solution of equation (1), being a characteristic number for a periodic solution of integral
order, was recently studied covering various aspects of convergence, and integral equations for the
second solution, using expansions in the �� ordinary and modified Mathieu functions, reproduced
Rayleigh 's formula for diffraction of electromagnetic waves (Hille, 1923).

In the case of an elliptical membrane, the solution is expressed in terms of the product of ordinary and
modified Mathieu functions. The zeros of the modified Mathieu functions determine the vibrational
angular frequencies and confocal nodal ellipses, while the modified Mathieu functions define a system
of confocal nodal hyperbolas. When the eccentricity of the bounded ellipse tends to zero, the nodal
ellipse tends to nodal circles, and the nodal hyperbola tends to nodal radii (Cesari, 1963).

THEORETICAL FRAMEWORK

For the work we have linear differential equations of the form
��
��

= ��(��)��, where ��(��) is a real, non-
singular square matrix and �� = (��1, ��2, … , ��) an n-dimensional vector. Linear differential equations that
appear naturally in the study of nonlinear equations, �̇� = ��(��, ��) where ��(��, ��) =
(��1(��, ��), ��2(��, ��), … , ��(��, ��)) is an n-dimensional vector function of �� y ��. If �� it is not a function of time,
then an expansion around fixed points, defined by the equation ��(��) = 0, normally gives equations with
constant matrix B, and the study of these linear equations provides important information about the
stability of those fixed points (Ross, 1995).

Alternatively, if the nonlinear system admits a periodic solution ��(��), ��(��) = ��(�� + ��), for any �� and all
T greater than zero, then an expansion of ��(��), leads to an equation with the matrix ��(��), T-periodic in
time.

Solution in terms of a fundamental matrix: the solution with the initial condition ��(��0) = ��0 can be
expressed as ��(��) = ��(��)��(��0)−1��(��0) where ��(��) is any fundamental matrix. The matrix ��(��2, ��1) =
��(��2)��(��1)−1, which depends on two times ��1 and ��2 is called the propagation matrix , this matrix takes
the solution from one time ��1 to ��2.

Floquet–Lyapunov theorem: for the equation of �̇� = ��(��)��, where the elements of the matrix ��(��), are
functions of ��, �� −periodic and piecewise continuous, with a number of discontinuities in (−∞, +∞)
and integrable in each discontinuity, a fundamental matrix is expressed in the form ��(��) = ��(��)��
where ��(��) is a square, non-singular matrix, T periodic for everything �� and with continuous elements,
with continuous derivatives by parts and integrables, (Barrow, 1996).

Perturbation theory: by applying perturbation theory, approximate analytical solutions of equations of
various types are obtained, which involve a small parameter. Perturbation theory is a collection of
methods for obtaining approximate analytical solutions of equations involving a small parameter ��,
taking into account that most methods of modern physics contain applications of perturbation theory.

It is very common in applications, when a model of a physical, chemical, or biological system is studied,
we have the following equation, which involves a small parameter �� defined in an interval �� = (0; ��0):
��(��, ��) = 0 where ��is the real variable, it is called a perturbed solution. If we know the unperturbed
solution, that is, the equation ��(��, 0) = 0, the asymptotic analysis leads to constructing the approximate
analytical solution for �� > 0 small (Bickley, 1940).

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Fundamental theorem of perturbation theory: if ��0��0(��) + ��1��1(��) + ��2��2(��) + ⋯ + ��(��) +
0(��+1(��)) = 0 where ��(��) is an asymptotic sequence and the coefficients �� (�� = 0,1,2,3, … , ��) are
independent of ��, then ��0 = ��1 = ��2 = ⋯ = �� = 0.

Floquet theory: for systems with periodic coefficients,
��
��

= ��(��)��, ��(�� + ��) = ��(��) for all ��, where ��(��)
is a real, non-singular matrix, with elements that are T-periodic functions of ��. In general, solutions of
periodic linear systems cannot be expressed in terms of elementary functions, but linearity and
periodicity mean ��(��) that the behavior of a solution for all times can be deduced from the general
solution over a finite interval of length ��. This property means that the behavior of solutions when �� →
∞ can often be deduced from analytical approximations or numerical solutions (Jordan, 1999).

Periodic multidimensional systems: for the homogeneous linear equation of order n,
��
��

=
��(��)��, ��(�� + ��) = ��(��) for all ��, where ��(��) is a square matrix, whose elements are T-periodic functions,
assuming that this equation has �� linearly independent solutions {��1(��), ��2(��), ��3(��), … , ��(��)}, it is
possible to form the fundamental matrix ��(��) that satisfies the equation

��
��

= ��(��)��. The vectors of

the form ��(��) = ��(�� + ��) are also solutions of the original differential equation, and we have
��
��

=
��(��+��)

��(��+��)
= ��(�� + ��)��(�� + ��) = ��(��)��. Therefore, ��(��) it must be a linear combination of the ��(��), �� =

1,2, … , ��, that is ��(��) = ∑��
��=1 ��(��)�� for some constants ��.

These new solutions can be used to form the fundamental matrix ��(�� + ��), and we have ��(�� + ��) =
��(��)��, where ��is a matrix with elements �� , since �� ( ��(�� + ��)) =�� (��(��)) �� (��)y

�� ( ��(��)) ≠ 0, �� is non-singular.

Importance of eigenvalues and eigenvectors: If �� is an eigenvalue and �� the associated eigenvector,
�� = ��, then the solution ��(��) = ��(��)�� has the property ��(�� + ��) = ��(��), for all ��, since ��(�� + ��) =
��(��)�� = ��(��)�� = ��(��), for all ��.

The eigenvalues of the matrix ��are called characteristic multipliers or characteristic numbers of the
system; The eigenvalues of �� are independent of the choice of the fundamental matrix, therefore, they
are a property of the system, not of any particular solution, if the different fundamental matrices ��1(��)
and ��2(��) give rise to the matrices ��1 and ��2, since ��2(��) = ��1(��)��, for some constant C, We also ��2 =
��−1��1�� have ��2(�� + ��) = ��1(��)��1�� = ��2(��)��−1��1��.

Similarity: ��1 and ��2are similar and have the same eigenvalues, then it is convenient to write the
eigenvalues in the form �� = �� , �� where ��

��
it is made unique by choosing its imaginary part,

and which satisfy −�� < 0(��) ≤ ��, therefore, ��
��

=
1
��

��(��) or �� = ��|��| + ��(��), −�� <
�� (��) ≤ ��, where �� is the main branch of the natural logarithm. The ��

��
are called the characteristic

exponents. If �� it has n distinct eigenvalues �� , the equation has n solutions linearly of the form ��(��) =
��(��)�� where ��(��) is a periodic vector function of time, then ��(�� + ��) = ��(��)��−��(��) = ��(��), for
everything �� (Edwards, 2000).

METHODOLOGY

T We follow physical aspects that generate a mathematical model, in this case, two problems that lead
to Mathieu 's equation:

Hill's lunar theory: the equations of motion of two planets with mass ��1y ��2and the sun with mass ��,
moving under Newton's laws without considering all the other effects that influence them, with ��1 and
��2 being position vectors of the planets and �� the radius vector position of the sun. The force on the

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Sun, taking the gravitational constant equal to unity, allows us to arrive at the equation ��´´ + ��2�� +
��
��2 =

0. The function �� ≡ 0 is solution of this equation; However, you should consider the case �� ≈ 0and see
how it behaves ��. Therefore, it is assumed that ��, �� they are periodic functions of ��, and of ��, the
equation can be written ��´´ + �� = 0, where �� = ��2 +

��
��3 is a function (Hirsch, et al, 1974).

Problem of the pendulum with variable length: considering a pendulum of variable length A with a mass
��, and a support that moves vertically with displacement ��(��), its Cartesian coordinates are �� = ��,
�� = ��(��) + ��(��), in order to find the equation of motion, we derive the equations to obtain the kinetic
and potential energies, ��´ = ��, ��´ = ��´(��) − ��(��), which is obtained �� =

��
2

[(��´)2 + (��´)2];

consequently, �� =
��
2

[(��)´2 + ��2(��´)2 − 2��´��´��(��)], with potential energy �� = −��(�� + ��(��),

with support in the Lagrange equation,
��

��
(

��
��

) −
��
��

= −
��
��

, results ��´´ + (�� − ��´´)�� = 0, which is
similarly ��´´ + (�� − 2��(2��))�� = 0 called the Mathieu equation (Vvedensky, 1993; Jeffreys, 1924).

RESULTS AND DISCUSSIONS

Hill equation

Be the Equation of the form
��2��
��2 + [�� + ��(��)]�� = 0, where �� is a constant and ��(��) a function of period

��. A special case of this equation can be seen if ��(��) = 2��(2��)y is the �� = �� Mathieu equation. By
doing so �� = �̇� it is possible to write the Hill equation in a standard matrix form

��
��

= ��(��)��, �� = (��1 ��2 ), �� = (0 1 − �� − ��(��) 0 ) (3)

in (3) as ��(��) = 0, and the equation ��(��) = ��(��0)�� (∫
��

��0
��(��(��)��) shows that the

�� (��) is constant and then �� (��) = 1 and the product of the eigenvalues is equal to one.
The eigenvalues of ��are given by ��

2
− ��(��)�� + 1 = 0 then 2�� = ��(��) ± √��(��)2 − 4. Therefore, the

long-term behavior of the solutions is determined mainly by the single real number ��(��).

Let us consider two independent solutions that satisfy the initial conditions ��
1

(0) = 1, ��1̇(0) = 0,
��2(0) = 0, �̇�2(0) = 1, therefore, ��(0) = �� in addition ��(��) = ��1(��) + �̇�2(��), there are five different
cases depending on the values of ��(��).

��(��) > 2. The eigenvalues are positive, different, not equal to +1and satisfy 0 < ��1 < 1 < ��2.The
characteristic exponents are ±��, where �� = ��2 > 0and two linearly independent solutions are
��1(��) = ��−��1(��), ��2(��) = ��2(��), where ��(��)are periodic functions with period ��.

��(��) = 2. The eigenvalues are identical and equal to +1; therefore �� = 0. We see that the behavior of
the solutions depends on the number of ��independent eigenvectors:

The matrix �� has two linearly independent eigenvectors: there are two solutions with period �� and
��1(��) = ��1(��), ��2(��) = ��2(��),

where ��(��) are T-periodic functions.

The matrix �� has an eigenvector linearly: the two independent solutions are ��1(��) = ��1(��), ��2(��) =
��1(��) + ��2(��), where ��(��)are T-periodic functions. The first solution is limited. The amplitude of the
second solution grows linearly with ��, so that there is an stable solution and an unstable solution.

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|��(��)| < 2. The eigenvalues of �� are complex and are written �� = ��±�� , where 0 < �� < ��, with
characteristic exponents �� = ±��−1(

��(��)
2

), consequently the two independent solutions are ��1(��) =
��1(��), ��2(��) = ��−��2(��), where ��(��) they are T-periodic functions. In this case all solutions are
bounded for all times ��.

��(��) = −2. The eigenvalues are identical and equal to -1. Again the behavior of the solutions depends
on the number of independent eigenvectors of ��:

The matrix �� has two linearly independent eigenvectors: there are two solutions with period 2�� because
�� = ��, and the two independent solutions are ��1(��) = ��1(��), ��2(��) = ��2(��), also satisfy the periodicity
conditions ��1(�� + ��) = −��1(��), ��2(�� + ��) = −��2(��).

The matrix �� has only one linearly independent eigenvector: the two independent solutions are ��1(��) =
��1(��), ��2(��) = ��1(��) + ��2(��)

where ��(��)are the 2T-periodic functions. The first solution is limited. The amplitude of the second
solution grows linearly with ��, therefore, there is a stable solution and an unstable solution.

��(��) < −2. The eigenvalues are real, negative, different from and not equal to -1 and satisfy ��2 < −1 <
��1 < 0. The characteristic exponents are �� = ±��(−��2) + ��, and the two linearly independent solutions
are ��1(��) = ��−��1(��), ��2(��) = ��2(��), with ��1(��) decreasing and ��2(��) increasing when �� → ∞.

This form of stability classification allows us to conjecture that there is a strong distinction between
stable and unstable solutions; which is true, but only for long times, or formally �� → ∞. If we look at a
solution in finite time, the distinction is not so clear, because solutions are generally smooth, continuous
differential functions that depend on the parameters of the system. Let's say that, if ��(��) = −2, the
amplitude of a solution grows linearly with ��: yes ��(��) = −2 + ��2 (Feynman, 1965).

The Mathieu equation

To illustrate the behavior analyzed previously in Hill's equation, let Mathieu 's equation be ,
��2��
��2 +

(�� − 2��2��)�� = 0, where we set the parameter ��, let �� = 2, then analyzing how the solutions depend
on parameters ��we calculate ��(��). The Sturm-Liouville theory shows that for 0 ≤ �� < 2�� and periodic
2�� solutions exist �� at discrete values of ��, which depends on ��. These values are classified according
to the convention ��0(��) < ��1(��) < ��1(��) < ⋯ , �� ≠ 0, with ��the even solutions and �� the odd solutions
being recognized . We also consider solutions for other values of �� (Shingareva, 1995).

The points where ��(��) = 2 they give the eigenvalues ��0(��), {��2��(��), ��2��(��)}, �� = 1,2, …, corresponding
to the Mathieu functions of period ��, �� = ��0(��, ��), {��2��(��, ��), ��2��(��, ��)}. The points where ��(��) = −2
they give the eigenvalues {��2��+(��), ��2��+1(��)}, �� = 1, 2, 3, …, corresponding to the Mathieu functions of
period 2�� {��2��+1(��, ��), ��2��+1(��, ��)}.

The damped Mathieu equation

The effects of adding a small linear damped term to the Mathieu equation becomes
��2��
��2 + ��

��
��

(�� − 2��2��)�� = 0, �� ≥ 0. Suppose it ��is small, when �� = 0, then it is a previous case. For �� small
regions of instability start from the points �� = ��2on the axis ��. The damped term, ��

��
��

, reduces energy
of the system, therefore, the damped term competes with the term that drives resonance; However,
hopefully the presence of damping of the areas associated with unsteady motion will decrease. The
damped Mathieu equation has the canonical form

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��

��
(�� ) = ��(��)(�� ), where

��
��

= ��, ��(��) = (0 − 1 2��(2��) − �� − �� ) (4)

we see that the determinant of the monodromy matrix ��, in (4) is �� (��) = ��−��, the period is ��.
Letting ��(��) = 2��, the eigenvalues of ��, the characteristic multipliers, are the solutions of ��

2
− 2�� +

��−�� = 0, whence ��± = �� ± √��2 − ��−��. The stability condition is |��±| ≤ 1, and because ��+��− =
��−�� < 1 we see that it is possible that both multipliers are real and that the solutions are stable.

Stability boundaries: if �� > 0then 0 < ��− < ��+, therefore, the stability boundary is given by the

condition ��+ = 1, i.e., �� =
1

2
(1 + ��−��). If �� < 0then ��− < ��+ < 0, then the stability boundary is by

the condition ��− = −1, that is, �� = −
1

2
(1 + ��−��), it follows that the stability condition is |��(��)| < 1 +

��−��.

If ��(��) = 1 + ��−��then ��+ = 1, and ��− = ��−�� and the general theory shows that two independent
solutions are ��+(��), and ��−(��)��−��where ��±(��) are �� periodic functions of ��. If ��(��) = −(1 + ��−��) then
��− = −1and ��+ = −��−�� and two independent solutions are ��−(��), and ��+(��)��−�� , where ��±(��) are 2��
periodic functions of ��.

DISCUSSION

Approximate analytical solutions: Let us consider the construction of periodic approximate analytical
solutions of the damped Mathieu equation using perturbation theory. Finding the expansions using
perturbation theory for periodic solutions in the presence of damping is more complicated than when
there is no damping, except if �� ≈ 0 or �� ≈ 1(Naytch, 1973).

The case: ��0 = 1, ��0 = 1, ��1 = √1 − ��2 The analysis necessary to obtain solutions through perturbation
theory is algebraically tedious, so we describe the first terms of the expansions for particular cases.
Complete calculations can be done using computer algebra methods (Shingareva, 1995). If we assume
that ��y ��are small and initially sets �� = �� to derive an expansion with respect to the parameter ��. The
equation is

��2��
��2 + ��(��)�� = �� (2��2�� − ��

��
��

), �� = ��.

Now we write ��(��) and ��(��) as a power series in ��,

��(��) = ��0(��) + ��1(��)�� + ��2(��)��2 + ��3��3 + ⋯

��(��) = ��0 + ��1�� + ��2��2 + ��3��3 + ⋯,

to obtain the infinite set of equations:

��2��0
��2 + ��0��0 = 0,

��2��1
��2 + ��0��1 = −��1��0 + 2��0��2�� − ��

��0
��

��2��2
��2 + ��0��2 = −��2��0 − ��1��1 + 2��1��2�� − ��

��1
��

…

��2��
��2 + ��0�� = − ∑��

��=1 ��−�� + 2��−1��2�� − ��
��−1

��
.

LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción, Paraguay.

ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1648.

This method can be used if ��0 = 0 or ��0 = 1but fails if ��0 ≥ 2. In this analysis of perturbation theory we
consider the case �� = 1, and therefore, we find an approximation for ��−(��), the - periodic solution 2��.
The solution to the equation is ��0(��) = ��0�� + ��0��, where ��0(��), ��0(��) are constants that must be
determined. Substituting the previous equation on the right side of the equation for ��1we have, after

some simplifications,
��2��1
��2 + ��1 = [��0(1 − ��1) − ��0]�� + [��(1 + ��1) − ��0]�� + ��0��3�� +

��0��3��. This equation has periodic solutions only if the coefficients of �� and �� are zero:
��0(1 − ��1) − ��0 = 0, ��0(1 + ��1) − ��0 = 0.

In the undamped limit, which is the case �� = 0, the equations have two solutions:

��1 = −1, ��0 = 0 → �� = ��0�� takes us to ��1(��, ��)

��1 = 1, ��0 = 0 → �� = ��0�� takes us to ��1(��, ��)

where ��0 and ��0 are chosen to satisfy appropriate normalization conditions.

If �� > 0 homogeneous equations have a single solution only if the determinant of the coefficients is
zero, this means that ��1

2 = 1 − ��2 it leads to two solutions

��1 = −√1 − ��2, ��0 =
��0

1+√1−��2
,

��1 = √1 − ��2, ��0 =
��0

1+√1−��2
.

But how ��0 = 1 the equation appears, in terms of the original variable �� = ��, the expressions: �� =
1 ± √��2 − ��2or ��2 − (�� − 1)2 = ��2, which are the equation of a hyperbola. This analysis shows that yes
0 ≤ �� < ��and �� ≈ 1 all solutions are stable, so for �� small, solutions that are unstable when �� = 0
become stable when �� > 0.

We now return to the derivation of the expansion of the problem. The only solutions are obtained by
setting the coefficient of �� or ��. The solution that reduces to ��1(��, ��) when �� = 0 is obtained by
leaving fixed ��1 = −√1 − ��2, ��0 = 1and will guarantee that the term ��in does not exist ��(��), �� ≥ 1.
Similarly, the solution that reduces to ��1(��, ��) when �� = 0 is obtained by leaving fixed ��1 = √1 − ��2,
��0 = 1 and there is no term �� in ��(��), �� ≥ 1.

In the next problem we will find the second solution by setting ��0 = 1 and ��1 = √1 − ��2. The equation

for ��1(��) becomes
��2��1
��2 + ��1 = ��3�� + ��0��3��, ��0 =

��
1+√1−��2

, having as a solution ��1(��) = −
1
8

��3�� −
��0
8

��3�� + ��1�� for some constant ��1. If we substitute this solution for ��2(��), equating the coefficients

of ��and �� to zero, we obtain the equations for ��2 �� 1, whose solutions are ��1 = 0, ��2 = −
1
8
. The

solution of this equation is ��2(��) = ��2�� −
1

64
(��1 + 3��0)��3�� −

1
64

(��1��0 − 3��)��3�� +
1

192
(��5�� +

��0��5��).

If we substitute this solution into the equation for ��3 and by factoring the coefficients of ��and ��we

obtain the equations for ��2 and ��3 giving the solutions ��2 =
3��

64√1−��2(1+√1−��2)
, ��3 = −

1+2��2

64√1−��2
. Therefore,

for ��(��)you have ��(��) = �� +
��

1+√1−��2
−

��
8

(��3�� +
��3��

1+√1−��2
) + 0(��2).

In the case �� = 0, these series reduce the series for ��1(��) and ��1(��, ��), for the values �� = 1, �� = �� =
0,2, the first series gives ��(��) = 1,84.

LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción, Paraguay.

ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1649.

Proposition. To ��0 = 4 show that if ��1(��) periodic, we have the equation ��1
2 + ��2 = 0. Establish that

this perturbation method is invalid. Show more generally, that a similar result is obtained if ��0 = ��2, �� ≥
2.

Indeed. As ��0 = 4, by the perturbation method described above we have the following equation
��2��
��2 +

4�� = 0 with the solution ��0(��) = ��0��2�� + ��0��2�� and the derivative
��0(��)

��
= −2��0��2�� + 2��0��2��.

We substitute it into the following equation of the method, the result is obtained:

��2��1
��2 + 4��1 = −(2��0 + ��1��0)��2�� + (2��0 − ��1��0)��2�� −

��0��4�� + ��0��4�� + ��0 , ( 5 )

and since for ( 5 ) ��1(��) is a periodic function, then the system of equations

{��1��0 + 2��0 = 0 2��0 − ��1��0 = 0 , ( 6 )

we solve ( 6 ). The equations are consistent only if the determinant of the system is zero, that is,
��(��1 2�� 2�� −��1 ) = −(��1

2 + 4��2) = 0, whose only solution is �� = 0 = ��1, therefore the method does

not work. Now in general, for �� = ��2, with �� ≥ 2, the same is done and we have the equation:
��2��
��2 +

��2��0 = 0, with the solution ��0(��) = ��0�� (��) + ��0��(��)and the derivative
��0(��)

��
= −��0�� +

��0�� we replace it in the equation of the perturbation method we obtain:
��2��1
��2 + 4��1 =

−(��0 + ��1��0)�� + (��0 − ��1��)��) − ��0��(2�� + ��) + ��0 �� (−2�� + ��) + ��0
�� (2�� + ��) + ��0��(−2�� + ��),

and as the problem says, if ��1(��)it is a periodic function, then

{��1��0 + ��0 = 0 ��0 − ��1��0 = 0 , (7)

we solve the system (7), and we see that ��(��1 �� −��1 ) = −(��1
2 + ��2��2) = 0, whose only solution

is ��1 = 0 and �� = 0, therefore, it is shown that the method is invalid for �� ≥ 2.

Method rectification: ��0 = 4. This last problem shows that an application of perturbation theory fails
when �� ≥ 2, can be overcome by fixing �� = ��when ��0 = ��2. For that matter �� = 2, we look for a
solution that reduces to ��2(��, ��) when �� → 0. We will start with the general solution for ��0(��), ��0 = 4,
��0(��) = ��0��2�� + ��0��2��, where ��0, ��0 are constants that must be determined. We look for solutions
for ��(��), �� ≥ 1, that do not contain the term ��2��. If we substitute ��0(��) into the equation for ��1(��)we
find that the equation has a periodic solution only if ��1 = 0 and then ��1(��) =

��0
4

−
��0
12

��4�� + ��1��2�� −
��0
12

��4��.

This expression in the equation for ��2(��) y, setting the coefficients of ��2�� y ��2�� to zero we obtain
the following equations ��0 (��2 −

5
12

) + 2��0 = 0, (��2 −
1

12
) ��0 − 2��0 = 0 for ��2 y ��0. If �� = 0 the only

non-trivial solutions are ��2 =
5

12
, ��0 = 1, ��0 = 0 and ��2 = −

1
12

, ��0 = 0, ��0 = 1, and leads to ��2(��, ��) and

��2(��, ��), respectively. If �� > 0 we can eliminate ��0 and ��0 to give an equation relating ��2 to ��. If we
remember that �� = 4 + ��2��2and �� = ��2, we obtain the following equation for ��(��), 4��2 +

(�� − 4 −
5��2

12
) (�� − 4 +

��2

12
) = 0. When �� = 0 this equation is reduced to the curves �� = 4 −

��2

12
, �� = 4 +

5��2

12
, corresponding to ��2(��) and ��2(��), respectively. The equations have two solutions ��0 =

2��0
��2+

1
12

, ��2 =

LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción, Paraguay.

ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1650.

1
6

+
1
4

√1 − 64��2, ��2(��, ��) when �� = 0, ��0 =
2��0

��2+
5

12
, ��2 =

1
6

+
1
4

√1 − 64��2, ��2(��, ��) when �� = 0, which are

real if 0 ≤ �� ≤
1
8
. From now on we choose the first of these solutions and set ��0 = 1. By perturbation

theory, the solution is ��2(��) =
1

384
(��6�� + ��0��6��) + ��2��2�� −

��1
12

��4��, where ��1 and ��2 are unknown
constants. If we substitute ��2(��) into the equation for ��3(��) equating the coefficients of ��2�� and ��2��
to zero we obtain ��1 = ��3 = 0.

A periodic solution for ��3(��) can be found if we substitute into the equation for ��4(��), and we obtain the
following linear equations:

��4 + 2��2 = −
��2
36

−
19��2
144

, ��4��0 + 2 (��2 +
1

12
) ��2 =

��
36

−
��0

4608
−

��0��2
144

These equations can be solved and the process continue. The series obtained for ��(��) is �� = 4 +
��2

12
(2 + 3√1 − 64��2) − ��4 (

763
13824

+
128

9
��4 + ⋯ ) + 0(��6).

Note that the coefficient of ��4 has been expanded into powers of �� to simplify the representation and
to show that when �� = 0, ��(��).

Mathieu 's functions: are the solutions ��and periodic of 2�� Mathieu 's equation
��2��
��2 + (�� − 2��2��)�� =

0, where �� and �� are constants which is a Sturm-Liouville system with periodic boundary conditions and
has solutions only for particular values of the eigenvalue ��, and that depends on ��, only periodic
solutions are considered. The big difference between Mathieu 's equation and other equations defining
special functions is that the coefficient of �� is a periodic function of the independent variable, and is

the simplest case of the more general equation
��2��
��2 + (�� + ��(��))�� = 0, where ��(��) is a periodic function

of �� (Nayteh, 1995).

Mathieu functions is complex, particularly when we have to understand that ��and �� depend on all
eigenfunctions. More details of the study with Maple can be found in Kreyszig (1994); Richards (2002).
The variable �� is a parameter that we need to get the behavior of both eigenvalues and both
eigenfunctions as a function ��. In this special case there are �� = 0 periodic solutions only if �� = ��2, �� =
0,1,2,3, … and these solutions are 1, ��, ��2��, ….(even solutions), ��, ��2��, …(odd solutions).

Mathieu functions corresponding to values of ��, when �� → ∞ denoted as follows: ��0(��, ��), ��1(��, ��),
��2(��, ��), … (even solutions), ��1(��, ��), ��2(��, ��), …, (odd solutions).

If �� ≠ 0 each of these eigenfunctions has a different eigenvalue. For each eigenvalue there exists at
most one period solution �� o 2��, and each pair {��(��, ��), ��(��, ��)} has �� zeros in the interval −�� < �� ≤
��. These solutions can be unique in several ways, one of these ways is by choosing the coefficient of
�� en ��(��, ��) and the coefficient of �� en ��(��, ��). The eigenvalue associated with the even
solutions, ��(��, ��), is denoted by ��(��), �� = 0,1,2,3, …, and the eigenvalue associated with the odd
solutions, ��(��, ��), as ��(��), �� = 0,1,2,3, …, Table 1.

Table 1

Mathieu functions

Function Period Parity environment
�� = 0

Parity
environment�� =

��
2

Eigenvalues

��2��(��, ��)
��2��+1(��, ��)

��
2��

Pair Pair
Odd

��2��(��),�� = 0,1, …
��2��+1(��),�� = 0,1, …

LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción, Paraguay.

ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1651.

��2��(��, ��)
��2��−1(��, ��)

��
2��

Odd Odd
Pair

��2��(��),�� = 0,1, …
��2��−1(��),�� = 0,1, …

Source: self made.

Mathieu functions

Mathieu 's equation is a Sturm-Liouville system
��

��
(��(��)

��
��

) + (��(��) + ��(��))�� = 0, with ��(��) = 1, ��(��) =
−2��2��, ��(��) = 1, �� = �� and with periodic boundary conditions.

Considering the space ��2, the set of eigenfunctions {��(��, ��)}is {��(��, ��)} complete on the interval
−�� ≤ �� ≤ ��. Additionally, each of the sets {��(��, ��)} y {��(��, ��)} is complete in 0 ≤ �� ≤ �� and each of
the sets {��2��(��, ��)} y {��2��+1(��, ��)}, {��2��(��, ��)} y {��2��+1(��, ��)} is complete in the interval 0 ≤ �� ≤ ��/2.

Mathieu functions are orthogonal ∫
��

−��
��(��, ��)��(��, ��)�� = ℎ��,

∫
��

−��
��(��, ��)��(��, ��)�� = 0, the normalization constant ℎ�� = ��.

The eigenvalues are ordered ��0(��) < ��1(��) < ��1(��) < ��2(��) < ��2(��) < ⋯ (�� ≠ 0).

It is convenient to divide eigenvalues into three types: eigenvalues with values much smaller than 2��,
eigenvalues whose values are much larger than , 2�� and eigenvalues that are around 2��.

For large eigenvalues , for which �� − 2��2�� is always positive, we have ��(��) → ��(��) → ��2 when �� →
0. For �� fixed and �� large, the difference between �� and �� , is exponentially small, ��(��) − ��(��) =
0 (

��

��−1) when �� → ∞.

Large ��eigenvalues can be approximated by the following ��(��) ≅ ��(��)~��2 +
��2

2��2 + ⋯series . Yes �� ≫
1, �� ≪ ��, therefore �� − 2��2�� it changes sign, ��+1(��) and ��(��) they are close.

For |��| small values, the eingenvalues can be expanded as a power series in ��, which can be found using
perturbation theory, but a clear view of the relationship between the ��(��) and the ��(��), is seen in the
following figure, in which the first twelve values eigenvalues are shown for 0 < �� < 35. The order of the

curves in this figure can be understood if we note that ��0 =
��2

2
+ ��(��4), for �� ≥ 1, ��(0) = ��(0) = ��2

and for �� > 1, ��(��) > ��(��), and for �� ≫ 1, ��(��) ≅ ��+1(��).

If �� < 0the change of variables �� →
��
2

− �� changes the sign of in ��2�� Mathieu 's equation, then the
following relationship holds ��2��(−��) = ��2��(��): , ��2��(−��) = ��2��(��), ��2��+1(−��) = ��2��+1(��), and

��2��(��, −��) = (−1)��2�� (
��
2

− ��, ��),

��2��+1(��, −��) = (−1)��2��+1 (
��
2

− ��, ��),

��2��+1(��, −��) = (−1)��2��+1 (
��
2

− ��, ��),

��2��(��, −��) = (−1)��−1��2�� (
��
2

− ��, ��).

Eigenfunctions with eigenvalues �� > 2�� They oscillate uniformly. However, there is a change in the
relationship between the even and odd functions, if �� ≫ 2��, ��(��) ≅ ��(��), while if �� ≪ 2��, ��+1(��) ≅
��(��).

LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades, Asunción, Paraguay.

ISSN en línea: 2789-3855, diciembre, 2024, Volumen V, Número 6 p 1653.

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