1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 4 471

an=^8 aU^0 (−^1 )

n+ 1

π^2 ( 2 n− 1 )^2

,bn=0.

23.

Y′′

Y =

2 V

k

ψ′

ψ.Thefunctionφ(x−Vt)cancels from both sides.

25.φn(−Vt)=T 0 exp(λ^2 nkt/ 2 )bn,t>0,

φn(x)=T 1 exp(λ^2 nkx/ 2 V)bn,x>0,

where

∑∞

n= 1

bnsin(λny)=1, 0<y<b.

27.φ(x−ct)=e−c(x−ct)/k=e(c^2 t−cx)/k.Thegivencsatisfies c^2 =iωk,

soφ(x−ct)=eiωt−(^1 +i)px=e−pxei(ωt−px). Now form^12 (φ(x−ct)+

φ(x−ct))=e−pxcos(ωt−px)and so forth.

29. Differentiate and substitute.
    31.φ(^2 )−φ(^4 )+λ^2 φ=0,
    φ( 0 )=0, φ(a)=0,
    φ′′( 0 )=0, φ′′(a)=0.

33.λn=

nπ

a

√

1 +

(nπ

a

) 2

∼=nπ

a

.

Chapter 4

Section 4.1

1.f+d=0.

3.Y(y)=Asinh(πy),A= 1 /sinh(π ).

5.v(r)=aln(r)+b.

7. ∂∂ux=∂v∂rcos(θ )−∂v∂θsinr(θ ),
   ∂u
   ∂y=

∂v

∂rsin(θ )+

∂v

∂θ

cos(θ )

r.

9. a.

∂^2 u

∂x^2 +

∂^2 u

∂y^2 =0,^0 <x<a,0<y<b,

u( 0 ,y)=0, u(a,y)=0, 0 <y<b,

u(x, 0 )=f(x), u(x,b)=f(x),0<x<a.