1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

476 Answers to Odd-Numbered Exercises

c.u(x,y)=

∑∞

1

ansin(nπx)exp(−n^2 π^2 y),

an= 2

∫ 1

0

f(x)sin(nπx)dx.

7.X′′/X=−λ^2 ,T′′/T=−λ^2 /( 1 +λ^2 ).

Chapter 4 Miscellaneous Exercises

1.u(x,y)=

∑∞

1

bn

sinh(λn(a−x))

sinh(λna) sin(λny),

λn=nπ/b,bn= 2 ( 1 −cos(nπ ))/nπ.

3.u(x,y)=1.Notethat0isaneigenvalue.

5.u(x,y)=

∑∞

n= 1

ansinh(λnx)+bnsinh(λn(a−x))

sinh(λna) cos(λny),

λn=( 2 n− 1 )π/ 2 b,an=bn= 4 (− 1 )n+^1 /π( 2 n− 1 ).

7.u(x,y)=w(x,y)+w(y,x),where

w(x,y)=

∑∞

n= 1

bnsinhsinh(λn(λ(a−y))

na)

sin(λnx),

λn=nπ/a,bn=

8 h

n^2 π^2 sin

(nπ

2

)

.

9.u(x,y)=

∫∞

0

A(λ)sinh(λ(b−y))

sinh(λb)

cos(λx)dλ,A(λ)=2sin(λa)/λπ.

11.u(x,y)=

∫∞

0

A(λ)cos(λx)e−λydλ,A(λ)= 2 α/π

(

α^2 +λ^2

)

.

13.u(x,y)=−π^1 tan−^1

(

x−x′

y

)∣∣

∣∣

∞

−∞

=π^1

[

π

2 −

(

−π 2

)]

.

15.u(r,θ)=a 0 +

∑∞

n= 1

(r

c

)n(

ancos(nθ)+bnsin(nθ)

)

,

a 0 =

1

2 ,an=0,bn=

1 −cos(nπ)

nπ.

17. Same form as Exercise 15, buta 0 = 2 /π,
    an= 2 ( 1 +cos(nπ ))/( 1 −n^2 ),bn=0(anda 1 =0).
    19.u(r,θ)=(ln(r)−ln(b))/(ln(a)−ln(b)).