method of integrating factors

method of integrating factors

The

method of integrating factors

is in principle a means for solving

ordinary differential equations

of first .  It has not great practical significance, but is theoretically important.

Let us consider a

differential equation

solved for the

derivative

y

′

of the unknown

function

and write the equation in the form

X

⁢

(

x

,

y

)

⁢

d

⁢

x

+

Y

⁢

(

x

,

y

)

⁢

d

⁢

y

=

0

.

(1)

We assume that the functions

X

and

Y

have

continuous

partial derivatives

in a region

R

of

ℝ

2

.

If there is a solution of (1) which may be expressed in the form

f

⁢

(

x

,

y

)

=

C

with

f

having continuous partial derivatives in

R

and with

C

an arbitrary

constant

, then it’s not difficult to see that such an

f

satisfies

the linear

partial differential equation

X

⁢

∂

⁡

f

∂

⁡

y

-

Y

⁢

∂

⁡

f

∂

⁡

x

=

0

.

(2)

Conversely

, every non-constant solution

f

of (2) gives also a solution

f

⁢

(

x

,

y

)

=

C

of (1).  Thus, solving (1) and solving (2) are

equivalent

(

http://planetmath.org/Equivalent3

) tasks.

It’s straightforward to show that if

f

0

⁢

(

x

,

y

)

is a non-constant solution of the equation (2), then all solutions of this equation are

F

⁢

(

f

0

⁢

(

x

,

y

)

)

where

F

is a freely chosen function with (mostly) continuous derivative.

The connection of the equations (1) and (2) may be presented also in another form.  Suppose that

f

⁢

(

x

,

y

)

=

C

is any solution of (1).  Then (2) implies the proportion equation

f

x

′

X

=

f

y

′

Y

.

If we denote the common value of these two ratios by

μ

⁢

(

x

,

y

)

=

μ

,  then we have

f

x

′

=

μ

⁢

X

,

f

y

′

=

μ

⁢

Y

.

This gives to the

differential

of the function

f

the expression

d

⁢

f

⁢

(

x

,

y

)

=

μ

⁢

(

x

,

y

)

⁢

(

X

⁢

(

x

,

y

)

⁢

d

⁢

x

+

Y

⁢

(

x

,

y

)

⁢

d

⁢

y

)

.

We see that

μ

⁢

(

x

,

y

)

is the

integrating factor

or

Euler multiplicator

of the given differential equation (1), i.e. the left hand side of (1) turns, when multiplied by

μ

⁢

(

x

,

y

)

,  to an exact differential (

http://planetmath.org/ExactDifferentialForm

).

Conversely, any integrating factor

μ

of (1), i.e. such that

μ

⁢

X

⁢

d

⁢

x

+

μ

⁢

Y

⁢

d

⁢

y

is the differential of some function

f

, is easily seen to determine the solutions of the form

f

⁢

(

x

,

y

)

=

C

of (1).  Altogether, solving the differential equation (1) is equivalent with finding an integrating factor of the equation.

When an integrating factor

μ

of (1) is available, the solution function

f

can be gotten from the

line integral

f

(

x

,

y

)

=

:

∫

P

0

P

[

μ

(

x

,

y

)

X

(

x

,

y

)

d

x

+

μ

(

x

,

y

)

Y

(

x

,

y

)

d

y

]

along any curve

γ

connecting an arbitrarily chosen point

P

0

=

(

x

0

,

y

0

)

and the point

P

=

(

x

,

y

)

in the region

R

.

Note.

In general, it’s very hard to find a suitable integrating factor.  One special case where such can be found, is that

X

and

Y

are homogeneous functions of same degree (

http://planetmath.org/HomogeneousFunction

): then the expression

1

x

⁢

X

+

y

⁢

Y

is an integrating factor.

Example.

In the differential equation

(

x

4

+

y

4

)

⁢

d

⁢

x

-

x

⁢

y

3

⁢

d

⁢

y

=

0

we see that

X

=

:

x

4

+

y

4

and

Y

=

:

-

x

y

3

both define a homogeneous function of degree (

http://planetmath.org/HomogeneousFunction

) 4.  Thus we have the integrating factor

μ

=

:

1

x

5

+

x

⁢

y

4

-

x

⁢

y

4

=

1

x

5

,  and the left hand side of the equation

(

1

x

+

y

4

x

5

)

⁢

d

⁢

x

-

y

3

x

4

⁢

d

⁢

y

=

0

is an exact differential.  We can integrate it along the broken line, first from

(

1

,

0

)

to

(

x

,

0

)

and then still to

(

x

,

y

)

,  obtaining

f

(

x

,

y

)

=

:

∫

1

x

(

1

x

+

0

4

x

5

)

d

x

-

∫

0

y

y

3

⁢

d

⁢

y

x

4

=

ln

|

x

|

-

y

4

4

⁢

x

4

.

So the

general solution

of the given differential equation is

ln

⁡

|

x

|

-

y

4

4

⁢

x

4

=

C

.

References

1

E. Lindelöf:

Differentiali- ja integralilasku III 1

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title

method of integrating factors

Canonical name

MethodOfIntegratingFactors

Date of creation

2013-03-22 16:31:48

Last modified on

2013-03-22 16:31:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

21

Author

pahio (2872)

Entry type

Topic

Classification

msc 35-00

Classification

msc 34-00

Related topic

ErnstLindelof

Defines

integrating factor

Defines

Euler multiplicator

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