Miscellaneous Exercises 355
both directly and by assuming that bothu(r)and the constant function 1
haveBesselseriesontheinterval0<r<a:u(r)=∑∞
n= 1CnJ 0 (λnr), 0 <r<a,1 =
∑∞
n= 1cnJ 0 (λnr), 0 <r<a.
(
Hint:^1 rdrd(
rdJ^0 dr(λr))
=−λ^2 J 0 (λr).)
6.Suppose thatw(x,t)andv(y,t)are solutions of the partial differential
equations
∂^2 w
∂x^2=^1
k∂w
∂t, ∂
(^2) v
∂y^2
=^1
k∂v
∂t.
Show thatu(x,y,t)=w(x,t)v(y,t)satisfies the two-dimensional heat
equation
∂^2 u
∂x^2 +∂^2 u
∂y^2 =1
k∂u
∂t.7.Use the idea of Exercise 6 to solve the problem stated in Exercise 1.
8.Letw(x,y)andv(z,t)satisfy the equations
∂^2 w
∂x^2+∂
(^2) w
∂y^2
= 0 , ∂
(^2) v
∂z^2
=^1
c^2∂^2 v
∂t^2.
Show that the productu(x,y,z,t)=w(x,y)v(z,t)satisfies the three-
dimensional wave equation
∂^2 u
∂x^2 +∂^2 u
∂y^2 +∂^2 u
∂z^2 =1
c^2∂^2 u
∂t^2.9.Find the product solutions of the equation
1
r∂
∂r(
r∂u
∂r)
=^1
k∂u
∂t, 0 <r, 0 <t,that are bounded asr→ 0 +and asr→∞.10.Show that the boundary value problem
(
( 1 −x^2 )φ′
)′
=−f(x), − 1 <x< 1 ,
φ(x)bounded atx=± 1 ,