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 ofnhomogeneous equations in thencoefficientsλr. The condition

that not all theλrbe zero is that the determinant of the coefficients of the

λrbe zero, that is,

∣ ∣ ∣ ∣ ∣ ∣ ∣

y 1 y 2 ··· yn

y

′

1

y

′

2

··· y

′

n

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

y

(n−1)

1

y

(n−1)

2

··· y

(n−1)

n

∣ ∣ ∣ ∣ ∣ ∣ ∣

=0

for all values ofx, which proves the theorem.

This determinant is known as the Wronskian of thenfunctionsyrand is

denoted byW(y 1 ,y 2 ,...,yn), which can be abbreviated toWnorWwhere

there is no risk of confusion. After transposition,Wncan be expressed in

column vector notation as follows:

Wn=W(y 1 ,y 2 ,...,yn)=

∣

∣

CC

′

C

′′

···C

(n−1)

∣

∣

where

C=

[

y 1 y 2 ···yn

]T

. (4.7.2)

IfWn= 0, identically thenfunctions are linearly independent.

Theorem 4.25. Ift=t(x),

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

n

W(y 1 ,y 2 ,...,yn).

Proof.

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

∣

∣(tC)(tC)′(tC)′′···(tC)(n−1)

∣

∣

=

∣

∣K

1 K 2 K 3 ···Kn

∣

∣,

where

Kj=(tC)

(j−1)

=D

j− 1

(tC),D=

d

dx

.

Recall the Leibnitz formula for the (j−1)th derivative of a product and

perform the following column operations:

K

′

j

=Kj+t

j− 1

∑

s=1

(

j− 1

s

)

D

s

(

1

t

)

Kj−s,j=n, n− 1 ,..., 3. 2.

=t

j− 1

∑

s=0

(

j− 1

s

)

D

s

(

1

t

)

Kj−s

=t

j− 1

∑

s=0

(

j− 1

s

)

D

s

(

1

t

)

D

j− 1 −s

(tC)

=tD

(j−1)

(C)

=tC

(j−1)

.