The Chemistry Maths Book, Second Edition

13.8 Exercises 389

The functionsj

l

(x)are called spherical Bessel functions of order l; the functions η

l

are the spherical Neumann functions. They satisfy the differential equation

x

2

y′′ 1 + 12 xy′ 1 + 1 [x

2

1 − 1 l(l 1 + 1 1)]y 1 = 10 (13.60)

These functions often occur in conjunction with the Legendre polynomials when a

physical system is formulated in spherical polar coordinates.

0 Exercise 27

13.8 Exercises

Section 13.2

Use the power-series method to solve the equations:

1.y′ 1 − 13 x

2

y 1 = 10

2.(1 1 − 1 x)y′ 1 − 1 y 1 = 10. Confirm the solution can be expressed asy 1 = 1 a 2 (1 1 − 1 x)when|x| 1 < 11.

3.y′′ 1 − 19 y 1 = 10. Confirm that the solution can be expressed asy 1 = 1 ae

3 x

1 + 1 be

− 3 x

.

4.(1 1 − 1 x

2

)y′′ 1 − 12 xy′ 1 + 12 y 1 = 10 (the Legendre equation (13.13) forl 1 = 11 ).

Show that the solution can be written as when|x| 1 < 11.

5.y′′ 1 − 1 xy 1 = 10 (Airy equation).

Section 13.3

For each of the following, find and solve the indicial equation

6.x

2

y′′ 1 + 13 xy′ 1 + 1 y 1 = 10 7.x

2

y′′ 1 + 1 xy′ 1 + 1 (x

2

1 − 1 n

2

)y 1 = 1 0 (Bessel equation)

8.xy′′ 1 + 1 (1 1 − 12 x)y′ 1 + 1 (x 1 − 1 1)y 1 = 10 9.x

2

y′′ 1 + 16 xy′ 1 + 1 (6 1 − 1 x

2

)y 1 = 10

10. (i)Find the general solution of the Euler–Cauchy equationx

2

y′′ 1 + 1 b

0

xy′ 1 + 1 c

0

y 1 = 10 for

distinct indicial roots,r

1

1 ≠ 1 r

2

. (ii)Show that for a double initial root r, the general

solution isy 1 = 1 (a 1 + 1 bln x)x

r

.

Solve the differential equations:

11. 12.x

2

y′′ 1 − 1 xy′ 1 + 1 y 1 = 10

13. (i)Solve the Bessel equationx

2

y′′ 1 + 1 xy′ 1 + 1 (x

2

1 − )y 1 = 10 for indicial rootr 1 = 1 − 122

(see Example 13.4 forr 1 = 1122 ). (ii)Confirm that the solution can be written as

14. (i)Use the expansion method to find a particular solutiony

1

(x)of

xy′′ 1 + 1 (1 1 − 12 x)y′ 1 + 1 (x 1 − 1 1)y 1 = 10.

(ii)confirm thaty

2

(x) 1 = 1 y

1

(x) 1 ln 1 xis a second solution.

15.Find the general solution ofxy′′ 1 + 12 y′ 1 + 14 xy 1 = 10. Assume that there is no logarithmic term

in the solution.

Section 13.4

16.Show that the polynomialP

l

(x)is a solution of the Legendre equation (13.13) for

(i)l 1 = 12 and (ii)l 1 = 15.

yx

a

x

x

a

x

()=+= +cos sinxaJ x bJ x() ()

−

01

12 12

1

4

xy xy y

2 1

2

1

2

0

′′

−

′

+=

yaxa

xx

x

=+ +

−

• 
  





























10

1

2

1

1

ln