The Chemistry Maths Book, Second Edition

374 Chapter 13Second-order differential equations. Some special functions

3 Roots differ by an integer

One solution has the form (13.4),

y

1

(x) 1 = 1 x

r

1

(a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-) (13.12a)

and the second solution is

y

2

(x) 1 = 1 ky

1

(x) 1 ln 1 x 1 + 1 x

r

2

(A

0

1 + 1 A

1

x 1 + 1 A

2

x

2

1 +1-) (13.12b)

in whichr

1

1 > 1 r

2

and the constant k may be zero.

0 Exercises 10–12

Many of the particular solutions of second-order differential equations that are of

interest in the physical sciences are generally series of type (13.4), without a logarithmic

term.

EXAMPLE 13.4The Bessel equationx

2

y′′ 1 + 1 xy′ 1 + 1 (x

2

1 − 1 n

2

)y 1 = 10 forn 1 = 1 ± 122.

By Example 13.3(ii), the indicial roots arer 1 = 1 ± 122 and the solutions are nominally

of type 3 , equations (13.12). In the present case, however, there is no logarithmic

term (see Exercise 13), and we show here that the particular solution with indicial

parameterr 1 = 1 + 122 is the Bessel function

We have

Then

and

=xaxa axa axa a+ +× + +× + +×

12

10 2

2

13

3

2

223 34 45()()(

44

4

x )+















xy xy x y x mm x x

mm

m

22

1

4

12 2

′′+ ′+− = + + 1













+

=

() ()

00

∞

∑

′=+

()

, ′′=−

( )

=

−

=

−

∑∑

y m ax y m ax

m

m

m

m

m

m

0

1

2

12

0

2

1

4

∞∞

332

y x ax ax

m

m

m

m

m

m

==

==

+

∑∑

12

00

12

∞∞

Jx

x

x

12

2

()= sin

π