aarjaneiro/planetmath huggingface dataset:
second order ordinary differential equation

second order ordinary differential equation

A

second order ordinary differential equation

F

⁢

(

x

,

y

,

d

⁢

y

d

⁢

x

,

d

2

⁢

y

d

⁢

x

2

)

=

0

can often be written in the form

d

2

⁢

y

d

⁢

x

2

=

f

⁢

(

x

,

y

,

d

⁢

y

d

⁢

x

)

.

(1)

By its

general solution

one means a

function

x

↦

y

=

y

⁢

(

x

)

which is at least on an

interval

twice

differentiable

and

satisfies

y

′′

⁢

(

x

)

≡

f

⁢

(

x

,

y

⁢

(

x

)

,

y

′

⁢

(

x

)

)

.

By setting

d

⁢

y

d

⁢

x

:=

z

,  one has

d

2

⁢

y

d

⁢

x

2

=

d

⁢

z

d

⁢

x

,  and the equation (1) reads

d

⁢

z

d

⁢

x

=

f

⁢

(

x

,

y

,

z

)

.  It is easy to see that solving (1) is

equivalent

(

http://planetmath.org/Equivalent3

) with solving the system of simultaneous

first order

(

http://planetmath.org/ODE

)

differential equations

{

d

⁢

y

d

⁢

x

=

z

,

d

⁢

z

d

⁢

x

=

f

⁢

(

x

,

y

,

z

)

,

(2)

the so-called

normal system

of (1).

The system (2) is a special case of the general

normal system of second order

, which has the form

{

d

⁢

y

d

⁢

x

=

φ

⁢

(

x

,

y

,

z

)

,

d

⁢

z

d

⁢

x

=

ψ

⁢

(

x

,

y

,

z

)

,

(3)

where

y

and

z

are unknown functions of the variable

x

.  The

existence theorem

of the solution

{

y

=

y

⁢

(

x

)

,

z

=

z

⁢

(

x

)

(4)

is as follows; cf. the Picard–Lindelöf

theorem

(

http://planetmath.org/PicardsTheorem2

).

Theorem.

If the functions

φ

and

ψ

are

continuous

and have continuous

partial derivatives

with respect to

y

and

z

in a neighbourhood of a point

(

x

0

,

y

0

,

z

0

)

,  then the normal system (3) has one and (as long as

|

x

-

x

0

|

does not exceed a certain ) only one solution (4)
which satisfies the

initial conditions

y

⁢

(

x

0

)

=

y

0

,

z

⁢

(

x

0

)

=

z

0

.  The functions (4) are continuously differentiable in a neighbourhood of

x

0

.

References

1

E. Lindelöf:

Differentiali- ja integralilasku III 1

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title

second order ordinary differential equation

Canonical name

SecondOrderOrdinaryDifferentialEquation

Date of creation

2013-03-22 18:35:39

Last modified on

2013-03-22 18:35:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Topic

Classification

msc 34A05

Defines

normal system

Defines

normal system of second order

Generated on Fri Feb 9 20:39:59 2018 by

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singular solution

singular solution

Let the

general solution

y

of the

differential equation

F

⁢

(

x

,

y

,

d

⁢

x

d

⁢

y

)

=

0

(1)

be given by the equation

G

⁢

(

x

,

y

,

C

)

=

0

,

(2)

the so-called

general integral

of (1); herein

C

means an arbitrary .

Suppose that the family (2) of the

integral curves

of (1) has an

envelope

.  In any point of the envelope, the tangent line is also the tangent of one integral curve.  Thus in such a point, all the three values

x

,

y

,

d

⁢

y

d

⁢

x

are same for the envelope curve and the integral curve.  But these values satisfy the equation (1).  Accordingly, this equation is satisfied also by the abscissa, ordinate and slope of the envelope.  This means that the envelope is an integral curve of the differential equation (1).  Because the envelope does not belong to the family (2), it cannot be obtained from the equation (2) with any value of

C

.

A solution of the differential equation (1) which cannot be obtained from the general integral, is called a

singular solution

of equation.

An envelope of the integral curves means always a singular solution, but sometimes a singular solution is only the locus of the

singular points

of the curves (e.g. for the equation

3

⁢

y

′

⁢

y

=

1

).

Note.

If the family (2) has an envelope, it may be found by eliminating

C

from the pair

G

⁢

(

x

,

y

,

C

)

=

0

,

G

C

′

⁢

(

x

,

y

,

C

)

=

0

.

Example.

Find the singular solution of the equation

y

2

(

1

+

y

′

)

2

=

R

2

.

(3)

We can separate the variables, getting

d

⁢

x

=

±

y

⁢

d

⁢

y

R

2

-

y

2

.

Integrating this yields

(

x

-

C

)

2

+

y

2

=

R

2

,

which represents the family of circles with radius

R

and centres on the

x

-axis.  The family has as envelope the double line

y

=

±

R

;  the functions

y

↦

±

R

satisfy also the differental equation (3) and thus are singular solutions of it.

Title

singular solution

Canonical name

SingularSolution

Date of creation

2013-03-22 18:05:58

Last modified on

2013-03-22 18:05:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 34A05

Synonym

singular solution of differential equation

Related topic

DeterminingEnvelope

Related topic

ClairautsEquation

Related topic

SeparationOfVariables

Defines

general integral

Generated on Fri Feb 9 20:40:04 2018 by

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solutions of ordinary differential equation

solutions of ordinary differential equation

Let us consider the

ordinary differential equation

F

⁢

(

x

,

y

,

y

′

,

y

′′

,

…

,

y

(

n

)

)

=

0

(1)

of order

n

.

The

general solution

of (1) is a function

x

↦

y

=

φ

⁢

(

x

,

C

1

,

C

2

,

…

,

C

n

)

satisfying the following conditions:

a)

y

depends on

n

arbitrary constants

C

1

,

C

2

,

…

,

C

n

.

b)

y

satisfies (1) with all values of

C

1

,

C

2

,

…

,

C

n

c) If there are given the

initial conditions

y

=

y

0

,

y

′

=

y

1

,

y

′′

=

y

2

,

…

,

y

(

n

-

1

)

=

y

n

-

1

when

x

=

x

0

,

then one can chose the values of

C

1

,

C

2

,

…

,

C

n

such that

y

=

φ

⁢

(

x

,

C

1

,

C

2

,

…

,

C

n

)

fulfils those conditions (supposing that

x

0

,

y

0

,

y

1

,

y

2

,

…

,

y

n

-

1

belong to the region where the conditions for the existence of the solution are valid).

Each function which is obtained from the general solution by giving certain concrete values for

C

1

,

C

2

,

…

,

C

n

,  is called a

particular solution

of (1).

Title

solutions of ordinary differential equation

Canonical name

SolutionsOfOrdinaryDifferentialEquation

Date of creation

2013-03-22 16:32:16

Last modified on

2013-03-22 16:32:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 34A05

Related topic

DerivativesOfSolutionOfFirstOrderODE

Defines

general solution

Defines

particular solution

Generated on Fri Feb 9 20:40:06 2018 by

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cycle

cycle

Let

x

˙

=

f

⁢

(

x

)

be an

autonomous

ordinary differential equation

defined by the vector field

f

:

V

→

V

then

x

⁢

(

t

)

∈

V

a solution of the system is a

cycle

(or

periodic solution

)

if it is a closed solution which is not an

equilibrium point

. The

period

of a cycle is the smallest positive

T

such that

x

⁢

(

t

)

=

x

⁢

(

t

+

T

)

.

Let

ϕ

t

⁢

(

x

)

be the flow defined by the above ODE and

d

be the metric of

V

then:

A cycle,

Γ

, is a

stable cycle

if for all

ϵ

>

0

there exists a neighborhood

U

of

Γ

such that for all

x

∈

U

,

d

⁢

(

ϕ

t

⁢

(

x

)

,

Γ

)

<

ϵ

.

A cycle,

Γ

, is

unstable cycle

if it is not a stable cycle.

A cycle,

Γ

, is

asymptotically stable cycle

if for all

x

∈

U

where

U

is a neighborhood of

Γ

,

lim

t

→

∞

⁡

d

⁢

(

ϕ

t

⁢

(

x

)

,

Γ

)

=

0

.

[

PL

]

example:

Let

x

˙

=

-

y

y

˙

=

x

then the above autonomous ordinary differential equations with initial value condition

(

x

⁢

(

0

)

,

y

⁢

(

0

)

)

=

(

1

,

0

)

has a solution which is a stable cycle. Namely the solution defined by

x

⁢

(

t

)

=

cos

⁡

t

y

⁢

(

t

)

=

sin

⁡

t

which has a period of

2

⁢

π

.

References

PL

Perko, Lawrence:

Differential Equations

and

Dynamical Systems

(Third Edition)

. Springer, New York, 2001.

Title

cycle

Canonical name

Cycle12

Date of creation

2013-03-22 15:00:51

Last modified on

2013-03-22 15:00:51

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Definition

Classification

msc 34A12

Classification

msc 34C07

Synonym

periodic solution

Synonym

stable periodic solution

Synonym

unstable periodic solution

Synonym

asymptotically stable periodic solution

Defines

period

Defines

stable cycle

Defines

unstable cycle

Defines

asymptotically stable cycle

Generated on Fri Feb 9 20:40:23 2018 by

LaTeXML
derivatives of solution of first order ODE

derivatives of solution of first order ODE

Suppose that

f

is a

continuously differentiable function

defined on an

open subset

E

of

ℝ

2

, i.e. it has on

E

the

continuous

partial derivatives

f

x

′

⁢

(

x

,

y

)

and

f

y

′

⁢

(

x

,

y

)

.

If

y

⁢

(

x

)

is a solution of the

ordinary differential equation

d

⁢

y

d

⁢

x

=

f

⁢

(

x

,

y

)

,

(1)

then we have

y

′

⁢

(

x

)

=

f

⁢

(

x

,

y

⁢

(

x

)

)

,

(2)

y

′′

⁢

(

x

)

=

f

x

′

⁢

(

x

,

y

⁢

(

x

)

)

+

f

y

′

⁢

(

x

,

y

⁢

(

x

)

)

⁢

y

′

⁢

(

x

)

(3)

(see the

http://planetmath.org/node/2798

general

chain rule

).  Thus there exists on

E

the

second derivative

y

′′

⁢

(

x

)

which is also continuous.  More generally, we can infer the

Theorem.

If

f

⁢

(

x

,

y

)

has in

E

the continuous partial derivatives up to the order

n

, then any solution

y

⁢

(

x

)

of the

differential equation

(1) has on

E

the continuous

derivatives

y

(

i

)

⁢

(

x

)

up to the order (

http://planetmath.org/OrderOfDerivative

)

n

+

1

.

Note 1.

The derivatives

y

(

i

)

⁢

(

x

)

are got from the equation (1) via succesive

differentiations

.  Two first ones are (2) and (3), and the next two ones, with a simpler notation:

y

′′′

=

f

x

⁢

x

′′

+

2

⁢

f

x

⁢

y

′′

⁢

y

′

+

f

y

⁢

y

′′

⁢

y

′

⁣

2

+

f

y

′

⁢

y

′′

,

y

(

4

)

=

f

x

⁢

x

⁢

x

′′′

+

3

⁢

f

x

⁢

x

⁢

y

′′′

⁢

y

′

+

3

⁢

f

x

⁢

y

⁢

y

′′′

⁢

y

′

⁣

2

+

f

y

⁢

y

⁢

y

′′′

⁢

y

′

⁣

3

+

3

⁢

f

x

⁢

y

′′

⁢

y

′′

+

3

⁢

f

y

⁢

y

′′

⁢

y

′

⁢

y

′′

+

f

y

′

⁢

y

′′′

Note 2.

It follows from (3) that the curve

f

x

′

⁢

(

x

,

y

)

+

f

y

′

⁢

(

x

,

y

)

⁢

f

⁢

(

x

,

y

)

=

0

(4)

is the locus of the

inflexion points

of the integral curves of (1), or more exactly, the locus of the points where the integral curves have with their

tangents

a contact of order (

http://planetmath.org/OrderOfContact

) more than one.  The curve (4) is also the locus of the points of tangency of the integral curves and their isoclines.

References

1

E. Lindelöf:

Differentiali- ja integralilasku III 1

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title

derivatives of solution of first order ODE

Canonical name

DerivativesOfSolutionOfFirstOrderODE

Date of creation

2013-03-22 18:59:14

Last modified on

2013-03-22 18:59:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 34A12

Classification

msc 34-00

Related topic

SolutionsOfOrdinaryDifferentialEquation

Related topic

InflexionPoint

Generated on Fri Feb 9 20:40:25 2018 by

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Grammian determinant

Grammian determinant

The Grammian determinant provides a necessary and

sufficient

method of determining whether a set of

continuous functions

f

1

,

f

2

,

…

,

f

n

is

linearly independent

on an

interval

I

=

[

a

,

b

]

with respect to the

inner product

⟨

f

i

|

f

j

⟩

=

∫

I

f

i

⁢

f

j

It is defined as:

G

⁢

(

f

1

,

f

2

,

…

,

f

n

)

=

|

∫

I

(

f

1

)

2

∫

I

f

1

⁢

f

2

⋯

∫

I

f

1

⁢

f

n

∫

I

f

2

⁢

f

1

∫

I

(

f

2

)

2

⋯

∫

I

f

2

⁢

f

n

⋮

⋮

⋱

⋮

∫

I

f

n

⁢

f

1

∫

I

f

n

⁢

f

2

⋯

∫

I

(

f

n

)

2

|

If the

functions

are

continuous

on

I

, then

G

=

0

if and only if the set of functions is linearly dependent. Note that the Grammian determinant is a special case of the more general

Gram determinant

.

Title

Grammian determinant

Canonical name

GrammianDeterminant

Date of creation

2013-03-22 17:37:33

Last modified on

2013-03-22 17:37:33

Owner

slider142 (78)

Last modified by

slider142 (78)

Numerical id

6

Author

slider142 (78)

Entry type

Definition

Classification

msc 34A12

Related topic

WronskianDeterminant

Related topic

GramDeterminant

Generated on Fri Feb 9 20:40:30 2018 by

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limit cycle

limit cycle

Let

x

˙

=

f

⁢

(

x

)

be a planar

autonomous

ordinary differential equation

and

Γ

be a

periodic solution

of the system. If the

α

-limit set (

http://planetmath.org/OmegaLimitSet

) or the

ω

-limit set (

http://planetmath.org/OmegaLimitSet

) of a solution with initial value not on

Γ

and the respective limit set is

Γ

then

Γ

is a

limit cycle

. In simpler terms a limit cycle is an isolated periodic solution of the system.

A limit cycle,

Γ

, is a

stable limit cycle

(or

ω

-limit cycle

)

if

Γ

is the

ω

-limit set of all solutions in some neighborhood of

Γ

.

A limit cycle,

Γ

, is a

unstable limit cycle

(or

α

-limit cycle

)

if

Γ

is the

α

-limit set of all solutions in some neighborhood of

Γ

.

[

PL

]

References

PL

Perko, Lawrence:

Differential Equations

and

Dynamical Systems

(Third Edition)

. Springer, New York, 2001.

Title

limit cycle

Canonical name

LimitCycle

Date of creation

2013-03-22 15:00:54

Last modified on

2013-03-22 15:00:54

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

9

Author

Daume (40)

Entry type

Definition

Classification

msc 34A12

Classification

msc 34C07

Synonym

ω

-limit cycle

Synonym

α

-limit cycle

Related topic

OmegaLimitSet

Defines

stable limit cycle

Defines

unstable limit cycle

Generated on Fri Feb 9 20:40:32 2018 by

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maximal interval of existence of ordinary differential equations

maximal interval of existence of ordinary differential equations

Let

E

⊂

W

where

W

is a

normed vector space

,

f

∈

C

1

⁢

(

E

)

is a

continuous

differentiable map

f

:

E

→

W

. Furthermore consider the

ordinary differential equation

x

˙

=

f

⁢

(

x

)

with the

initial condition

x

⁢

(

0

)

=

x

0

.

For all

x

0

∈

E

there exists a unique solution

x

:

I

→

E

where

I

=

[

-

a

,

a

]

, which also satify the initial condition of the

initial value problem

. Then there exists a maximal

interval

of existence

J

=

(

α

,

β

)

such that

I

⊂

J

and there exists a unique solution

x

:

J

→

E

.

Title

maximal interval of existence of ordinary differential equations

Canonical name

MaximalIntervalOfExistenceOfOrdinaryDifferentialEquations

Date of creation

2013-03-22 13:37:06

Last modified on

2013-03-22 13:37:06

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

8

Author

Daume (40)

Entry type

Theorem

Classification

msc 34A12

Classification

msc 35-00

Classification

msc 34-00

Generated on Fri Feb 9 20:40:33 2018 by

LaTeXML
Peano’s theorem

Peano’s theorem

If

d

⁢

y

d

⁢

x

=

f

⁢

(

x

,

y

)

(1)

is an

ordinary differential equation

where

f

⁢

(

x

,

y

)

is

continuous

on a planar domain

E

, then (

1

) has at least one integral
curve for each

(

x

0

,

y

0

)

of

E

.

[

KF

,

T

]

References

KF

Kolmogorov, A.N. & Fomin, S.V.

,

Introductory

Real Analysis

, Translated & Edited by Richard A. Silverman

, Dover Publications, Inc. New York, 1970.

T

Teschl, Gerald

,

http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html

http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html, 2004.

Title

Peano’s theorem

Canonical name

PeanosTheorem

Date of creation

2013-03-22 15:20:21

Last modified on

2013-03-22 15:20:21

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Theorem

Classification

msc 34A12

Generated on Fri Feb 9 20:40:35 2018 by

LaTeXML
Picard’s theorem

Picard’s theorem

Theorem 1

(Picard’s theorem

[

KF

]

).

Let

E

be an open subset of

R

2

and a

continuous function

f

⁢

(

x

,

y

)

defined as

f

:

E

→

R

. If

(

x

0

,

y

0

)

∈

E

and

f

satisfies

the

Lipschitz condition

in the variable

y

in

E

:

|

f

⁢

(

x

,

y

)

-

f

⁢

(

x

,

y

1

)

|

≤

M

⁢

|

y

-

y

1

|

where

M

is a

constant

. Then the

ordinary differential equation

defined as

d

⁢

y

d

⁢

x

=

f

⁢

(

x

,

y

)

with the

initial condition

y

⁢

(

x

0

)

=

y

0

has a unique solution

y

⁢

(

x

)

on some

interval

|

x

-

x

0

|

≤

δ

.

The above

theorem

is also named the

Picard-Lindelöf theorem

and can be generalized to a system of

first order

ordinary differential equations

Theorem 2

(generalization of Picard’s theorem

[

KF

]

).

Let

E

be an open subset of

R

n

+

1

and a continuous function

f

⁢

(

x

,

y

1

,

…

,

y

n

)

defined as

f

=

(

f

1

,

…

,

f

n

)

:

E

→

R

n

. If

(

t

0

,

y

10

,

…

,

y

n

⁢

0

)

∈

E

and

f

satisfies the Lipschitz condition in the variable

y

1

,

…

,

y

n

in

E

:

|

f

i

⁢

(

x

,

y

1

,

…

,

y

n

)

-

f

i

⁢

(

x

,

y

1

′

⁢

…

,

y

n

′

)

|

≤

M

⁢

max

1

≤

j

≤

n

⁡

|

y

j

-

y

j

′

|

where

M

is a constant. Then the

system of ordinary differential equation

defined as

d

⁢

y

1

d

⁢

x

=

f

1

⁢

(

x

,

y

1

,

…

,

y

n

)

⋮

d

⁢

y

n

d

⁢

x

=

f

n

⁢

(

x

,

y

1

,

…

,

y

n

)

with the initial condition

y

1

⁢

(

x

0

)

=

y

10

,

…

,

y

n

⁢

(

x

0

)

=

y

n

⁢

0

has a unique solution

y

1

⁢

(

x

)

,

…

,

y

n

⁢

(

x

)

on some interval

|

x

-

x

0

|

≤

δ

.

see also:

•

the entry

existence and uniqueness of solution of ordinary differential equations

References

KF

Kolmogorov, A.N. & Fomin, S.V.: Introductory

Real Analysis

, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.

Title

Picard’s theorem

Canonical name

PicardsTheorem

Date of creation

2013-03-22 14:59:57

Last modified on

2013-03-22 14:59:57

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Theorem

Classification

msc 34A12

Synonym

Picard-Lindelöf theorem

Related topic

ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations

Generated on Fri Feb 9 20:40:38 2018 by

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using Laplace transform to solve initial value problems

using Laplace transform to solve initial value problems

Since the

Laplace transforms

of the derivatives of

f

⁢

(

t

)

are
polynomials in the

transform parameter

s

(see table of Laplace
transforms), forming the Laplace transform of a

linear differential
equation

with constant coefficients and

initial conditions

at

t

=

0

yields generally a simple equation
(

image equation

(

http://planetmath.org/imageequation

)) for solving the transformed

function

F

⁢

(

s

)

.  Since the initial conditions can be taken into consideration instantly, one needs not to determine the

general solution

of the

differential equation

.

For example, transforming the equation

f

′′

(

t

)

+

2

f

′

(

t

)

+

f

(

t

)

=

e

-

t

(

f

(

0

)

=

0

,

f

′

(

0

)

=

1

)

gives

[

s

2

⁢

F

⁢

(

s

)

-

s

⁢

f

⁢

(

0

)

-

f

′

⁢

(

0

)

]

+

2

⁢

[

s

⁢

F

⁢

(

s

)

-

f

⁢

(

0

)

]

+

F

⁢

(

s

)

=

1

s

+

1

,

i.e.

(

s

2

+

2

⁢

s

+

1

)

⁢

F

⁢

(

s

)

=

1

+

1

s

+

1

,

whence

F

⁢

(

s

)

=

1

(

s

+

1

)

2

+

1

(

s

+

1

)

3

.

Taking the inverse Laplace transform produces the result

f

⁢

(

t

)

=

t

⁢

e

-

t

+

t

2

⁢

e

-

t

2

=

e

-

t

2

⁢

(

t

2

+

2

⁢

t

)

.

Title

using Laplace transform to solve

initial value problems

Canonical name

UsingLaplaceTransformToSolveInitialValueProblems

Date of creation

2015-05-29 15:21:45

Last modified on

2015-05-29 15:21:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Example

Classification

msc 34A12

Classification

msc 44A10

Related topic

TableOfLaplaceTransforms

Related topic

LaplaceTransform

Generated on Fri Feb 9 20:40:40 2018 by

LaTeXML
Fuchsian differential equation

Fuchsian differential equation

A

linear ordinary differential equation

is called Fuchsian if every

singular point

, including infinity is regular (

http://planetmath.org/FuchsianSingularity

). The form of such an equation is rather restricted. The coefficients of the equation must be rational functions.

Title

Fuchsian differential equation

Canonical name

FuchsianDifferentialEquation

Date of creation

2013-03-22 14:47:29

Last modified on

2013-03-22 14:47:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 34A25

Generated on Fri Feb 9 20:40:41 2018 by

LaTeXML
Fuchsian singularity

Fuchsian singularity

Suppose that

D

is an

open subset

of

ℂ

and that the

n

functions

c

k

:

D

→

ℂ

,

k

=

0

,

…

,

n

-

1

are

meromorphic

. Consider the

ordinary differential equation

d

n

⁢

w

d

⁢

z

n

+

∑

k

=

0

n

-

1

c

k

⁢

(

z

)

⁢

d

k

⁢

w

d

⁢

z

k

=

0

A point

p

∈

D

is said to be a

regular singular point

or a

Fuchsian singular point

of this equation if at least one of the functions

c

k

has a pole at

p

and, for every value of

k

between

0

and

n

, either

c

k

is

regular

at

p

or has a pole of order not greater than

n

-

k

.

If

p

is a Fuchsian singular point, then the

differential equation

may be rewritten as a system of first order equations

d

⁢

v

i

d

⁢

z

=

1

z

⁢

∑

j

=

1

n

b

i

⁢

j

⁢

(

z

)

⁢

v

j

⁢

(

z

)

in which the coefficient functions

b

i

⁢

j

are analytic at

z

. This fact helps explain the restiction on the orders of the poles of the

c

k

’s.

If an equation has a Fuchsian singularity, then the solution can be expressed as a Frobenius series in a

neighborhood

of this point.

A

singular point

of a differential equation which is not a regular singular point is known as an irregular singular point.

Examples

The

Bessel equation

w

′′

+

1

z

⁢

w

′

+

z

2

-

1

z

2

⁢

w

=

0

has a Fuchsian singularity at

z

=

0

since the coefficient of

w

′

has a pole of order

1

and the coefficient of

w

has a pole of order

2

.

On the other hand, the

Hamburger equation

w

′′

+

2

z

⁢

w

′

+

z

2

-

1

z

4

⁢

w

=

0

has an

irregular singularity

at

z

=

0

since the coefficient of

w

has a pole of order

4

.

Title

Fuchsian singularity

Canonical name

FuchsianSingularity

Date of creation

2013-03-22 14:47:26

Last modified on

2013-03-22 14:47:26

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

14

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 34A25

Synonym

Fuchsian singular point

Synonym

regular singular point

Synonym

regular singularity

Related topic

FrobeniusMethod

Defines

irregular singular point

Defines

irregular singularity

Defines

Hamburger equation

Generated on Fri Feb 9 20:40:43 2018 by

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