1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 1 451

b.sin(x)

x

=

∫∞

0

A(λ)cos(λx)dλ,whereA(λ)=

{ 1 , 0 <x<1,

0 , 1 <x.

5. a.A(λ)≡0,B(λ)=2sin(λπ )
   π( 1 −λ^2 )

;

b.A(λ)=^1 +cos(λπ )

π( 1 −λ^2 )

,B(λ)= sin(λπ )

π( 1 −λ^2 )

;

c.A(λ)=^2 (^1 π(+ 1 cos−λ(λπ )) (^2) ) ,B(λ)≡0.

7. Change variable fromxtoλwithx=λz.

Section 1.10

1.eαx= 2 sinhπ(απ )

(

1

2 α+

∑∞

n= 1

(− 1 )n

α^2 +n^2

(

αcos(nx)−nsin(nx)

)

)

.

3.f(x)=

∫∞

−∞

C(λ)eiλxdλ.

a.C(λ)=^1

2 π( 1 +iλ)

;b.C(λ)=^1 +e

−iλπ

2 π( 1 −λ^2 )

.

5. a. 1+

∑∞

n= 1

rncos(nx)=Re

∑∞

0

(

reix

)n

=Re^1

1 −reix

;

b.

∑∞

n= 1

sin(nx)

n!

=Im

∑∞

n= 1

einx

n!

=Im exp

(

eix

)

.

7. a.f(x)=2sin(x)
   x

;b.f(x)=^2

1 +x^2

.

Section 1.11

1.u(t)=A 0 +

∑∞

n= 1

Ancos(nt/ 2 )+Bnsin(nt/ 2 ),

A 0 =

1

2. 08 , An=

0. 4 /π

( 1. 04 −n^2 )^2 +( 0. 4 n)^2 ,

Bn=−

1

nπ

1. 04 −n^2

( 1. 04 −n^2 )^2 +( 0. 4 n)^2.

3.u(x)=

∑∞

n= 1

Bnsin(nπx/L),Bn=

8 Ksin(nπ/ 2 )

((nπ/L)^2 +γ^2 )n^2 π^2 ,

K=w/EI,γ^2 =T/EI.