variation of parameters

variation of parameters

The method of

variation of parameters

is a way of finding a particular
solution to a

nonhomogeneous

linear differential equation.

Suppose that we have an

n

th order linear

differential operator

L

⁢

[

y

]

:=

y

(

n

)

+

p

1

⁢

(

t

)

⁢

y

(

n

-

1

)

+

⋯

+

p

n

⁢

(

t

)

⁢

y

,

and a corresponding nonhomogeneous

differential equation

L

⁢

[

y

]

=

g

⁢

(

t

)

.

(1)

Suppose that we know a fundamental set of solutions

y

1

,

y

2

,

…

,

y

n

of the corresponding

homogeneous differential equation

L

⁢

[

y

c

]

=

0

. The

general solution

of the

homogeneous equation

is

y

c

⁢

(

t

)

=

c

1

⁢

y

1

⁢

(

t

)

+

c

2

⁢

y

2

⁢

(

t

)

+

⋯

+

c

n

⁢

y

n

⁢

(

t

)

,

where

c

1

,

c

2

,

…

,

c

n

are

constants

.
The general solution to the nonhomogeneous equation

L

⁢

[

y

]

=

g

⁢

(

t

)

is then

y

⁢

(

t

)

=

y

c

⁢

(

t

)

+

Y

⁢

(

t

)

,

where

Y

⁢

(

t

)

is a particular solution which

satisfies

L

⁢

[

Y

]

=

g

⁢

(

t

)

, and the
constants

c

1

,

c

2

,

…

,

c

n

are chosen to satisfy the appropriate

boundary conditions

or

initial conditions

.

The key step in using variation of parameters is to suppose that the
particular solution is given by

Y

⁢

(

t

)

=

u

1

⁢

(

t

)

⁢

y

1

⁢

(

t

)

+

u

2

⁢

(

t

)

⁢

y

2

⁢

(

t

)

+

⋯

+

u

n

⁢

(

t

)

⁢

y

n

⁢

(

t

)

,

(2)

where

u

1

⁢

(

t

)

,

u

2

⁢

(

t

)

,

…

,

u

n

⁢

(

t

)

are as yet to be determined

functions

(hence the name

variation of parameters

). To find
these

n

functions we need a set of

n

independent equations.
One

obvious

condition is that the proposed ansatz satisfies Eq.
(

1

). Many possible additional conditions are possible,
we choose the ones that make further calculations easier. Consider the
following set of

n

-

1

conditions

y

1

⁢

u

1

′

+

y

2

⁢

u

2

′

+

⋯

+

y

n

⁢

u

n

′

=

0

y

1

′

⁢

u

1

′

+

y

2

′

⁢

u

2

′

+

⋯

+

y

n

′

⁢

u

n

′

=

0

⋮

y

1

(

n

-

2

)

⁢

u

1

′

+

y

2

(

n

-

2

)

⁢

u

2

′

+

⋯

+

y

n

(

n

-

2

)

⁢

u

n

′

=

0

.

Now, substituting Eq. (

2

) into

L

⁢

[

Y

]

=

g

⁢

(

t

)

and using the
above conditions, we can get another equation

y

1

(

n

-

1

)

⁢

u

1

′

+

y

2

(

n

-

1

)

⁢

u

2

′

+

⋯

+

y

n

(

n

-

1

)

⁢

u

n

′

=

g

.

So we have a system of

n

equations for

u

1

′

,

u

2

′

,

…

,

u

n

′

which
we can solve using Cramer’s rule:

u

m

′

⁢

(

t

)

=

g

⁢

(

t

)

⁢

W

m

⁢

(

t

)

W

⁢

(

t

)

,

m

=

1

,

2

,

…

,

n

.

Such a solution always exists since the

Wronskian

W

=

W

⁢

(

y

1

,

y

2

,

…

,

y

n

)

of the system is nowhere zero, because the

y

1

,

y

2

,

…

,

y

n

form a
fundamental set of solutions. Lastly the term

W

m

is the Wronskian
determinant with the

m

th column replaced by the column

(

0

,

0

,

…

,

0

,

1

)

.

Finally the particular solution can be written explicitly as

Y

⁢

(

t

)

=

∑

m

=

1

n

y

m

⁢

(

t

)

⁢

∫

g

⁢

(

t

)

⁢

W

m

⁢

(

t

)

W

⁢

(

t

)

⁢

𝑑

t

.

References

1

W. E. Boyce and R. C. DiPrima.

Elementary

Differential Equations and

Boundary Value Problems

John Wiley & Sons, 6th edition, 1997.

Title

variation of parameters

Canonical

name

VariationOfParameters

Date of creation

2013-03-22 12:39:16

Last modified on

2013-03-22 12:39:16

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 34A30

Classification

msc 34A05

Synonym

variation of constants

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