Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 99

Hence,

W(ty 1 ,ty 2 ,...,tyn)=

∣

∣

(tC)(tC

′

)(tC

′′

)···(tC

(n−1)

)

∣

∣

=t

n

∣

∣CC′C′′···C(n−1)

∣

∣.

The theorem follows.

Exercise.Prove that

d

n

x

dyn

=

(−1)

n+1

W{y

′′

,(y

2

)

′′

,(y

3

)

′′

...(y

n− 1

)

′′

}

1!2!3!···(n−1)!(y

′

)

n(n+1)/ 2

,

wherey
′
=dy/dx,n≥2. (Mina)

4.7.2 The Derivatives of a Wronskian

The derivative ofWnwith respect tox, when evaluated in column vector

notation, consists of the sum ofndeterminants, only one of which has

distinct columns and is therefore nonzero. That determinant is the one

obtained by differentiating the last column:

W

′

n

=

∣

∣CC′C′′···C(n−3)C(n−2)C(n)

∣

∣.

Differentiating again,

W

′′

n=

∣

∣

CC

′

C

′′

···C

(n−3)

C

(n−1)

C

(n)

∣

∣

+

∣

∣

CC

′

C

′′

···C

(n−3)

C

(n−2)

C

(n+1)

∣

∣

, (4.7.3)

etc. There is no simple formula forW

(r)

n. In some detail,

W

′

n=

∣

∣

∣

∣

∣

∣

∣

∣

y 1 y

′

1 ··· y

(n−2)

1

y

(n)

1

y 2 y

′

2

··· y

(n−2)

2

y

(n)

2

..........................

yn y

′

n ··· y

(n−2)

n y

(n)

n

∣

∣

∣

∣

∣

∣

∣

∣

n

. (4.7.4)

The first (n−1) columns ofW

′

n are identical with the corresponding

columns ofWn. Hence, expandingW

′

nby elements from its last column,

W

′

n

=

n

∑

r=1

y

(n)

r

W

(n)

rn

. (4.7.5)

Each of the cofactors in the sum is itself a Wronskian of order (n−1):

W

(n)

rn

=(−1)

r+n

W(y 1 ,y 2 ,...,yr− 1 ,yr+1,...,yn). (4.7.6)

W

′

nis a cofactor ofWn+1:

W

′

n

=−W

(n+1)

n+1,n

. (4.7.7)

Repeated differentiation of a Wronskian of ordernis facilitated by adopting

the notation

Wijk...r=

∣

∣C(i)C(j)C(k)···C(r)

∣

∣