Miscellaneous Exercises 51
14.Show that the boundary value problem
d^2 u
dx^2+λ^2 u=f(x), 0 <x<a,
u( 0 )= 0 , u(a)= 0 ,
will have no solution or infinitely many solutions ifλis an eigenvalue of
d^2 u
dx^2 +λ(^2) u= 0 ,
u( 0 )= 0 , u(a)= 0.
Chapter Review
See the CD for review questions.
Miscellaneous Exercises
In Exercises 1–15, solve the given boundary value problem, supplying bound-
edness conditions where necessary.
1.d^2 u
dx^2 −γ(^2) u=0, 0 <x<a,
u( 0 )=T 0 , u(a)=T 1.
- d
(^2) u
dx^2 −r=0,^0 <x<a (ris constant),
u( 0 )=T 0 , du
dx
(a)=0.
3.
d^2 u
dx^2 =0,^0 <x<a,
u( 0 )=T 0 , dudx(a)=0.- d
(^2) u
dx^2
−γ^2 u=0, 0 <x<a,
du
dx(^0 )=0, u(a)=T^1.
5.^1
r
d
dr(
rdu
dr)
=−p,0<r<a,u(a)=0.