1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

454 Answers to Odd-Numbered Exercises

35.a 0 =

a

6 ,an=

2 a

n^2 π^2

(

cos

( 2 nπ

3

)

−cos

(nπ

3

))

.

37.a 0 =^58 ,an=n (^22) π 2

(

3cos

(

nπ

2

)

− 2 −cos(nπ)

)

.

39.a 0 =

1

2 ,an=

2

n^2 π^2

(

1 −cos(nπ)

)

.

41.a 0 =a

2

6 ,an=

− 2 a^2

n^2 π^2

(

1 +cos(nπ)

)

.

43.a 0 =

1

2 ,an=

− 1

nπ2sin

(nπ

2

)

.

45.bn=^1 +cos(nπ/^2 )−2cos(nπ)

nπ

.

47.bn=a

(2sin(nπ/ 2 )

n^2 π^2 −

cos(nπ)

nπ

)

.

49.bn=n^2 π

(

cos

(

nπ

4

)

−cos

(

3 nπ

4

))

.

51.bn= 2 nπ(^1 −e

kacos(nπ))

(a^2 k^2 +n^2 π^2 )

.

53.A(λ)=^2

π( 1 +λ^2 )

.

55.A(λ)=

2sin(λb)

πλ.

57.A(λ)=

2 ( 1 −cos(λ))

πλ^2.

59.B(λ)=^2 λ

π( 1 +λ^2 )

.

61.B(λ)=

2 ( 1 −cos(λb))

λπ.

63.B(λ)=

2 (λ−sin(λ))

λ^2 π.

65. The termancos(nx)+bnsin(nx)appears inSn,Sn+ 1 ,...,SN,andthus
    N+ 1 −ntimes inσN.


66. Use Eq. (13) of Section 7 and the identity in Exercise 66.


67. a. Usex=0; b.x= 1 /2; c.x=0.