Page 113 - Determinants and Their Applications in Mathematical Physics

98 4. Particular Determinants

Proof. Equation (4.7.1) together with its first (n − 1) derivatives form
a set of n homogeneous equations in the n coefficients λ r . The condition
that not all the λ r be zero is that the determinant of the coefficients of the
λ r be zero, that is,

y 1 y 2 ···
y n
y y ··· y

1 2 n =0

...........................
(n−1) (n−1)
y y (n−1)
1 2 ··· y n
for all values of x, which proves the theorem.
This determinant is known as the Wronskian of the n functions y r and is
denoted by W(y 1 ,y 2 ,...,y n ), which can be abbreviated to W n or W where
there is no risk of confusion. After transposition, W n can be expressed in
column vector notation as follows:

W n = W(y 1 ,y 2 ,...,y n )= CC C ··· C (n−1)

where

. (4.7.2)
T
C = y 1 y 2 ··· y n
If W n = 0, identically the n functions are linearly independent.
Theorem 4.25. If t = t(x),
W(ty 1 ,ty 2 ,...,ty n )= t W(y 1 ,y 2 ,...,y n ).
n
Proof.

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

= K 1 K 2 K 3 ··· K n ,

where
d
j−1
(j−1)
K j =(tC) = D (tC), D = .
dx
Recall the Leibnitz formula for the (j − 1)th derivative of a product and
perform the following column operations:
j−1
j − 1 1
K = K j + t D s K j−s , j = n, n − 1,..., 3.2.

j s t
s=1
j−1
j − 1 1
= t D s
s t K j−s
s=0
j−1
j − 1
1
= t D s D j−1−s (tC)
s t
s=0
= tD (j−1) (C)
= tC (j−1) .