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4.7 Wronskians 97

and where
φ =(m +1)φ m−1 , φ 0 = constant,
m
prove that

A = n(n − 1)A n−1 .
n
3. Prove that
1 b 12 xb 13 x 2 ··· ··· b 1,n+1 x n

−1 1 b 23 x b 2,n+1 x n−1

n ··· ···
1 a r x

n−2
−1 1 ··· ··· b 3,n+1 x ,
−1 1 =

r=1 ··· ··· ··· ···

−1 1

n+1
where
j−1

b ij = a r .
r=i
4. If
u u /2! u /3! u (4) /4! ···

u u u /2! u /3! ···

u u u /2! ··· ,

U n =
u u

···

··· ···
n
prove that
uU
n
U n+1 = u U n − . (Burgmeier)
n +1
4.7 Wronskians
4.7.1 Introduction
Let y r = y r (x), 1 ≤ r ≤ n, denote n functions each with derivatives of
orders up to (n − 1). These functions are said to be linearly dependent if
there exist coefficients λ r , independent of x and not all zero, such that
n

λ r y r = 0 (4.7.1)
r=1
for all values of x.
Theorem 4.24. The necessary condition that the functions y r be linearly
dependent is that
(i−1)
y =0

j
n
identically.