Wronskian determinant

Wronskian determinant

Given

functions

f

1

,

f

2

,

…

,

f

n

, then the

Wronskian determinant

(or simply the

Wronskian

)

W

⁢

(

f

1

,

f

2

,

f

3

,

…

,

f

n

)

is the

determinant

of the

square matrix

W

⁢

(

f

1

,

f

2

,

f

3

,

…

,

f

n

)

=

|

f

1

f

2

f

3

⋯

f

n

f

1

′

f

2

′

f

3

′

⋯

f

n

′

f

1

′′

f

2

′′

f

3

′′

⋯

f

n

′′

⋮

⋮

⋮

⋱

⋮

f

1

(

n

-

1

)

f

2

(

n

-

1

)

f

3

(

n

-

1

)

⋯

f

n

(

n

-

1

)

|

where

f

(

k

)

indicates the

k

th

derivative

of

f

(not

exponentiation

).

The Wronskian of a set of functions

F

is another function, which is zero over any interval where

F

is

linearly dependent

. Just as a set of vectors is said to be linearly dependent when there exists a non-trivial linear

relation

between them, a set of functions

{

f

1

,

f

2

,

f

3

,

…

,

f

n

}

is also said to be dependent over an interval

I

when there exists a non-trivial linear relation between them, i.e.,

a

1

⁢

f

1

⁢

(

t

)

+

a

2

⁢

f

2

⁢

(

t

)

+

⋯

+

a

n

⁢

f

n

⁢

(

t

)

=

0

for some

a

1

,

a

2

,

…

,

a

n

, not all zero, at any

t

∈

I

.

Therefore the Wronskian can be used to determine if functions are independent. This is useful in many situations. For example, if we wish to determine if two solutions of a

second-order

differential equation

are independent, we may use the Wronskian.

Examples

Consider the functions

x

2

,

x

, and

1

. Take the Wronskian:

W

=

|

x

2

x

1

2

⁢

x

1

0

2

0

0

|

=

-

2

Note that

W

is always non-zero, so these functions are independent everywhere. Consider, however,

x

2

and

x

:

W

=

|

x

2

x

2

⁢

x

1

|

=

x

2

-

2

⁢

x

2

=

-

x

2

Here

W

=

0

only when

x

=

0

. Therefore

x

2

and

x

are independent except at

x

=

0

.

Consider

2

⁢

x

2

+

3

,

x

2

, and

1

:

W

=

|

2

⁢

x

2

+

3

x

2

1

4

⁢

x

2

⁢

x

0

4

2

0

|

=

8

⁢

x

-

8

⁢

x

=

0

Here

W

is always zero, so these functions are always dependent. This is intuitively obvious, of course, since

2

⁢

x

2

+

3

=

2

⁢

(

x

2

)

+

3

⁢

(

1

)

Given

n

linearly independant functions

f

1

,

f

2

,

…

,

f

n

, we can use
the Wronskian to construct a linear differential equation whose solution space
is exactly the span of these functions. Namely, if

g

satisfies

the equation

W

⁢

(

f

1

,

f

2

,

f

3

,

…

,

f

n

,

g

)

=

0

,

then

g

=

a

1

⁢

f

1

⁢

(

t

)

+

a

2

⁢

f

2

⁢

(

t

)

+

⋯

+

a

n

⁢

f

n

⁢

(

t

)

for some choice of

a

1

,

a

2

,

…

,

a

n

.

As a simple illustration of this, let us consider polynomials of at most

second order

. Such a polynomial is a

linear combination

of

1

,

x

, and

x

2

. We
have

W

⁢

(

1

,

x

,

x

2

,

g

⁢

(

x

)

)

=

|

1

x

x

2

g

⁢

(

x

)

0

1

2

⁢

x

g

′

⁢

(

x

)

0

0

2

g

′′

⁢

(

x

)

0

0

0

g

′′′

⁢

(

x

)

|

=

2

⁢

g

′′′

⁢

(

x

)

.

Hence, the equation is

g

′′′

⁢

(

x

)

=

0

which indeed has exactly polynomials
of degree at most two as solutions.

Title

Wronskian determinant

Canonical name

WronskianDeterminant

Date of creation

2013-03-22 12:22:59

Last modified on

2013-03-22 12:22:59

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 34-00

Synonym

Wronskian

Related topic

GrammianDeterminant

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