1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Answers to Odd-Numbered Exercises 443

c.λ=±

nπ

a,n=^0 ,^1 ,^2 ,....

5.c=−a/2,c′=h−^1

μ

cosh

(μa

2

)

.

7.u(x)=T+c 1 cosh(γx)+c 2 sinh(γx),where

γ=

√

hC

κA

andc 1 =T 0 −T,c 2 =−κγsinh(γa)+hcosh(γa)

κγcosh(γa)+hsinh(γa)

c 1.

9.u(x)=T+H

(

1 −cosh(γx)−^1 −sinhcosh(γ(γa)a)sinh(γx)

)

,

whereH=I

(^2) R
hCandγ=

√

hC

κA.

11.u(y)=y(L−y)g/ 2 μ.

13.P=EI(nπ/L)^2 ,n= 1 , 2 ,....

15.u(x)=T+A

(

1 −cosh(γx)−^1 −cosh(γa)

sinh(γa)

sinh(γx)

)

,

A=g/κγ^2 ,andγ=

√

hC

κA

.

17.u(r)=c 1 ln(r/a)+c 2 , c 1 = h 0 h 1 (Ta −TW)/D, c 2 =[h 0 (κ/b+

h 1 ln(b/a))TW+(κ/a)h 1 Ta]/D,D=h 1 κ/a+h 0 κ/b+h 0 h 1 ln(b/a).

19.u(x)=

w 0

EI

(x 4

24 −

ax^3

6 +

a^2 x^2

2

)

.

Section 0.4

1. a.u′′+^1 ru′−u=0,r=0;

b.u′′−^2 x

1 −x^2

u′=0,x=±1;

c.u′′+cot(φ)u′−u=0,φ=0,±π,± 2 π,...;

d.u′′+^2

ρ

u′+λ^2 u=0,ρ=0.

3.u( 0 )bounded;u(ρ)=H

6 κ

(

c^2 −ρ^2

)

+Hc

3 h

+T.

5.u(ρ)=

1

ρ

(

c 1 cos(μρ)+c 2 sin(μρ)

)

,

u(ρ)≡0unlessμa=π, 2 π,.... The critical radius isa=π

μ

.