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560 CHAPTER 15 Special Functions and Eigenfunction Expansions

We obtain
π
2 cos(nψ)

b n = (ψ − ϕ)
π ψ
0
2 π cos(nψ)
+ dϕ
π 0 n
2 π cos(nψ)
− dψ
π 0 n
2 π
= cos(nψ)dϕ
nπ 0
2 π
= cos(nϕ − ne sin(ϕ))dϕ
nπ 0
2
= J n (ne).
n
The last line comes from Bessel’s integrals. The solution to Kepler’s problem is therefore
∞
2

ϕ = ψ + J n (ne)sin(nψ).
n
n=1
SECTION 15.3 PROBLEMS

a
c
1. Show that x J ν (bx ) is a solution of 10. Use the change of variables
1 du
2a − 1 a − ν c bu =

2 2 2
2 2 2c−2 u dx
y − y + b c x + y = 0.
x x 2
to transform the differential equation
dy 2 m
In each of Problems 2 through 9, write the general solu- dx + by = cx
x
a
tion of the differential equation in terms of x J ν (bx ) and
a
c
x J −ν (bx for appropriate a, b and c. into the differential equation
2
d u
m
− bcx u = 0.
dx 2

1 7

2. y + y + 1 + y = 0 Find the general solution of this differential equation
3x 144x 2
in terms of Bessel functions and use this to solve
1
4 the original differential equation. Assume that b is a
2
3. y + y + 4x − y = 0
x 9x 2 positive constant.
5
5
6

4. y − y + 64x + y = 0 In each of Problems 11 through 16, use the given change
x x 2
of variables to transform the differential equation into one
3
5
2 whose general solution can be written in terms of Bessel
5. y + y + 16x − y = 0
x 4x 2 functions. Use this to write the general solution of the
3 original differential equation.
4
6. y − y + 9x y = 0
x √
2
11. 4x y + 4xy + (x − 9)y = 0; z = x

7 175
4

2
7. y − y + 36x + y = 0 12. 4x y + 4xy + (9x − 36)y = 0; z = x 3/2
3

x 16x 2
2

13. 9x y + 9xy + (4x 2/3 − 16)y = 0; z = 2x 1/3

1 1

2
8. y + y − y = 0 14. 9x y − 27xy + (9x + 35)y = 0;u = yx −2
2

x 16x 2
2
2
15. 36x y − 12xy + (36x + 7)y = 0;u = yx −2/3

5 7
4
√

9. y + y + 81x + y = 0 16. 4x y + 8xy + (4x − 35)y = 0;u = y x
2
2

x 4x 2
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October 14, 2010 15:20 THM/NEIL Page-560 27410_15_ch15_p505-562