theory for separation of variables

theory for separation of variables

The

first order

(

http://planetmath.org/ODE

)

ordinary differential equation

where one can separate the

variables

has the form where

d

⁢

y

d

⁢

x

may be expressed as a

product

or a

quotient

of two

functions

(

http://planetmath.org/ProductOfFunctions

), one of which depends only on

x

and the other on

y

.  Such an equation may be written e.g. as

d

⁢

y

d

⁢

x

=

Y

⁢

(

y

)

X

⁢

(

x

)

or

d

⁢

x

d

⁢

y

=

X

⁢

(

x

)

Y

⁢

(

y

)

.

(1)

We notice first that if

Y

⁢

(

y

)

has real zeroes (

http://planetmath.org/ZeroOfAFunction

)

y

1

,

y

2

,

…

, then the equation (1) has the

constant

solutions

y

:=

y

1

,

y

:=

y

2

,

…

and thus the lines

y

=

y

1

,

y

=

y

2

,

…

are integral curves.  Similarly, if

X

⁢

(

x

)

has real zeroes

x

1

,

x

2

,

…

, one has to include the lines

y

=

y

1

,

y

=

y

2

,

…

to the integral curves.  All those lines the

x

⁢

y

-plane into the rectangular regions.  One can obtain other integral curves only inside such regions where the

derivative

d

⁢

y

d

⁢

x

attains real values.

Let

R

be such a region, defined by

a

<

x

<

b

,

c

<

y

<

d

,

and let us assume that the

X

⁢

(

x

)

and

Y

⁢

(

y

)

are real,

continuous

and distinct from zero in

R

.  We will show that any integral curve of the

differential equation

(1) is

accessible

by two

quadratures

.

Let

γ

be an integral curve passing through the point

(

x

0

,

y

0

)

of the region

R

.  By the above

assumptions

, the derivative

d

⁢

y

d

⁢

x

maintains its sign on the curve

γ

so long

γ

is inside

R

, which is true on a neighbourhood

N

of

x

0

,

contained

in

[

a

,

b

]

.  This implies that as

x

runs the

interval

N

,  it defines the ordinate

y

of

γ

uniquely as a

monotonic function

y

↦

y

⁢

(

x

)

which

satisfies

the equation (1):

y

′

⁢

(

x

)

=

Y

⁢

(

y

⁢

(

x

)

)

X

⁢

(

x

)

The last equation may be written

y

′

⁢

(

x

)

Y

⁢

(

y

⁢

(

x

)

)

=

1

X

⁢

(

x

)

.

(2)

Since

X

and

Y

don’t vanish in

R

, the

denominators

Y

⁢

(

y

⁢

(

x

)

)

and

X

⁢

(

x

)

are distinct from 0 on the interval

N

.  Therefore one can

integrate

both sides of (2) from

x

0

to an arbitrary value

x

on

N

, getting

∫

x

0

x

y

′

⁢

(

x

)

⁢

d

⁢

x

Y

⁢

(

y

⁢

(

x

)

)

=

∫

x

0

x

d

⁢

x

X

⁢

(

x

)

.

(3)

Because

y

=

y

⁢

(

x

)

is continuous and

monotonic

on the interval

N

, it can be taken as new variable of integration (

http://planetmath.org/SubstitutionForIntegration

) in the left hand side of (3):  substitute

y

⁢

(

x

)

:=

y

,

y

′

⁢

(

x

)

⁢

d

⁢

x

:=

d

⁢

y

and change the to

y

⁢

(

x

0

)

=

y

0

and

y

⁢

(

x

)

=

y

.

•

Accordingly, the equality

∫

y

0

y

d

⁢

y

Y

⁢

(

y

)

=

∫

x

0

x

d

⁢

x

X

⁢

(

x

)

(4)

is valid, meaning that if an integral curve of (1) passes through the point

(

x

0

,

y

0

)

, the integral curve is represented by the equation (4) as long as the curve is inside the region

R

.

•

Additionally, it is possible to justificate that if

(

x

0

,

y

0

)

is an interior point of a region

R

where

X

⁢

(

x

)

and

Y

⁢

(

y

)

are real, continuous and

≠

0

, then one and only one integral curve of (1) passes through this point, the curve is

regular

(

http://planetmath.org/RegularCurve

), and both

x

and

y

are monotonic on it.  N.B., the

Lipschitz condition

for the right hand side of (1) is not necessary for the justification.

•

When the point

(

x

0

,

y

0

)

changes in the region

R

, (4) gives a family of integral curves which cover the region once.  The equations of these curves may be unified to the form

∫

d

⁢

y

Y

⁢

(

y

)

=

∫

d

⁢

x

X

⁢

(

x

)

,

(5)

which thus the

general solution

of the differential equation (1) in

R

.  Hence one can speak of the

separation of variables

,

d

⁢

y

Y

⁢

(

y

)

=

d

⁢

x

X

⁢

(

x

)

,

(6)

and integration of both sides.

References

1

E. Lindelöf:

Differentiali- ja integralilasku III 1

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title

theory for separation of variables

Canonical

name

TheoryForSeparationOfVariables

Date of creation

2013-03-22 18:37:43

Last modified on

2013-03-22 18:37:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Topic

Classification

msc 34A09

Classification

msc 34A05

Related topic

InverseFunctionTheorem

Related topic

ODETypesReductibleToTheVariablesSeparableCase

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