Page 114 - Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 99

Hence,

)
W(ty 1 ,ty 2 ,...,ty n )= (tC)(tC )(tC ) ··· (tC (n−1)

= t CC C ··· C (n−1) .
n
The theorem follows.
Exercise. Prove that

2
d x (−1) n+1 W{y , (y ) , (y ) ... (y n−1
3
) }
n
= ,
n(n+1)/2
dy n 1!2!3! ··· (n − 1)!(y )
where y = dy/dx, n ≥ 2. (Mina)

4.7.2 The Derivatives of a Wronskian
The derivative of W n with respect to x, when evaluated in column vector
notation, consists of the sum of n determinants, only one of which has
distinct columns and is therefore nonzero. That determinant is the one
obtained by differentiating the last column:
(n−3) (n−2)

W = CC C ··· C C C (n) .

n
Differentiating again,
(n−3) (n−1)

W = CC C ··· C C C (n)

n
(n−3) (n−2)

+ CC C ··· C C C (n+1) , (4.7.3)

(r)
etc. There is no simple formula for W n . In some detail,
(n−2) (n)
y ··· y y
y 1
1 1

(n−2) 1
y ··· y y (n)
W = y 2 2 2 2 . (4.7.4)

n

(n−2)
..........................
y
(n)
y n
n ··· y n y n n
The first (n − 1) columns of W are identical with the corresponding
n

columns of W n . Hence, expanding W by elements from its last column,
n
n
(n) (n)

W = y W . (4.7.5)
n r rn
r=1
Each of the cofactors in the sum is itself a Wronskian of order (n − 1):
W (n) =(−1) r+n W(y 1 ,y 2 ,...,y r−1 ,y r+1 ,...,y n ). (4.7.6)
rn

W is a cofactor of W n+1 :
n
(n+1)

W = −W . (4.7.7)
n n+1,n
Repeated differentiation of a Wronskian of order n is facilitated by adopting
the notation
(i) (j) (k)
W ijk...r = C C C ··· C (r)