1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

42 Chapter 0 Ordinary Differential Equations

and identify the singular point(s).

a.^1

r

d

dr

(

rdu

dr

)

=u;

c. d

dφ

(

sin(φ)du

dφ

)

=sin(φ)u;

b. d

dx

((

1 −x^2

)du

dx

)

=0;

d.^1

ρ^2

d

dρ

(

ρ^2 du

dρ

)

=−λ^2 u.

2.The temperatureuinalargeobjecthavingaholeofradiuscin the middle

may be said to obey the equations

1

r

d

dr

(

rdu

dr

)

= 0 , r>c,

u(c)=T.

Solve the problem, adding the appropriate boundedness condition.

3.Compact kryptonite produces heat at a rate ofHcal/s cm^3 .Ifasphere

(radiusc) of this material transfers heat by convection to a surrounding

medium at temperatureT,thetemperatureu(ρ)in the sphere satisfies the

boundary value problem

1

ρ^2

d

dρ

(

ρ^2 du

dρ

)

=−H

κ

, 0 <ρ<c,

−κdu

dρ

(c)=h

(

u(c)−T

)

.

Supply the proper boundedness condition and solve. What is the tempera-

ture at the center of the sphere?

4.(Critical radius) The neutron fluxuin a sphere of uranium obeys the dif-

ferential equation

λ

3

1

ρ^2

d

dρ

(

ρ^2 dduρ

)

+(k− 1 )Au= 0

in the range 0<ρ<a,whereλis the effective distance traveled by a neu-

tron between collisions,Ais called the absorption cross section, andkis

the number of neutrons produced by a collision during fission. In addition,

the neutron flux at the boundary of the sphere is 0. Make the substitution

u=v/ρand 3(k− 1 )A/λ=μ^2 , and determine the differential equation

satisfied byv(ρ). See Section 0.1, Exercise 19.

5.Solve the equation found in Exercise 4 and then findu(ρ)that satisfies

the boundary value problem (with boundedness condition) stated in Ex-

ercise 4. For what radiusais the solution not identically 0?