The Chemistry Maths Book, Second Edition

13.3 The Frobenius method 371

Then

and this is zero if the coefficient of each power of xis separately zero:

a

m

1 + 1 (m 1 + 1 2) 1 (m 1 + 1 1)a

m+ 2

1 = 10

For the even values of m,

and for the odd values of m,

Therefore,

We recognize the two series in square brackets as the power-series expansions of

cos 1 xandsin 1 x. Therefore

y 1 = 1 a

0

1 cos 1 x 1 + 1 a

1

1 sin 1 x

wherea

0

anda

1

are arbitrary constants (see equation (12.22) withω 1 = 11 ).

0 Exercises 3 –5

13.3 The Frobenius method

The Frobenius method

2

is an extended power-series method that is used to solve

second-order linear equations that can be written in the form

y′′+ ′+= (13.3)

bx

x

y

cx

x

y

() ()

2

0

=−

!

• 
  

!

−

!

• 
  

















+−

!

• 
  

!

a

xxx

ax

xx

0

246

1

35

1

246 35

 −−

!

• 
  

















x

7

7



y a ax ax=+++ ax ax ax













++++











02

2

4

4

13

3

5

5





a

aa

a

aa

a

a

3

11

5

31

7

5

32 3 54 5 76

=−

×

=−

!

,=−

×

=+

!

,=−

×

=−

aa

1

7!

,

a

a

a

aa

a

aa

2

0

4

20

6

40

21 43 4 65 6

=−

×

,=−

×

=+

!

,=−

×

=−

!

,

=+++













=

+

∑

m

mm

m

am max

0

2

21

∞

()()

′′+= + + +

==

+

∑∑

yy ax m max

m

m

m

m

m

m

00

2

21

∞∞

()()

2

Georg Frobenius (1849–1917), German mathematician, is also known for his work in matrix algebra. In a

monograph in 1878 he organized the theory of matrices into the form that it has today.