1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

52 Chapter 0 Ordinary Differential Equations

6.

1

r

d

dr

(

r

du

dr

)

=0, a<r<b,

u(a)=T 0 , u(b)=T 1.

7.^1
ρ^2

d

dρ

(

ρ^2 du

dρ

)

=−H,0<ρ<a,

u(a)=T 0.

8.^1
r

d

dr

(

rdu

dr

)

=0, a<r<∞,

u(a)=T.

9. d

(^2) u
dx^2 −γ
(^2) (u−T)=0, 0 <x<a,
du
dx
( 0 )=0, u(a)=T 1.

10.

d^2 u

dx^2 −γ

(^2) u=0, 0 <x<∞,
u( 0 )=T.

11. d

(^2) u
dx^2 =γ
(^2) (u−T 0 ),0<x<∞,
u( 0 )=T.
12.
d
dx

(

x^3

du

dx

)

=−k, a<x<b (kis constant),

u(a)=0, u(b)=0(Note:0<a.)

13.In this problem,his the groundwater level between two trenches in

which water is held at constant levels. Solve forh(x). Note that the equa-

tion is nonlinear.

d

dx

(

hdh

dx

)

+e= 0 , 0 <x<a,

h( 0 )=h 0 , h(a)=h 1.

14.Solve foru(x).

d^4 u

dx^4 =w,^0 <x<a (wis constant),

u( 0 )= 0 , u(a)= 0 , d

(^2) u
dx^2
( 0 )= 0 , d
(^2) u
dx^2
(a)= 0.