464 Answers to Odd-Numbered Exercises
- Product solutions areφn(x)Tn(t),where
φn(x)=sin(λnx), Tn(t)=exp(
−kc^2 t/ 2)
×
{
sin(μnt)
cos(μnt),λn=nπ
a,μn=√
λ^2 nc^2 −^1
4k^2 c^4.- Product solutions areφn(x)Tn(t),where
φn(x)=sin(nπx
a)
,
Tn(t)=sin or cos(n (^2) π (^2) ct
a^2
)
.
Frequenciesn^2 π^2 c/a^2.- The general solution of the differential equation isφ(x)=Acos(λx)+
Bsin(λx)+Ccosh(λx)+Dsinh(λx). Boundary conditions atx=0re-
quireA=−C,B=−D;thoseatx=alead toC/D=−(cosh(λa)+
cos(λa))/(sinh(λa)−sin(λa))and 1+cos(λa)cosh(λa)=0. The first
eigenvalues areλ 1 = 1. 875 /a,λ 2 = 4. 693 /a, and the eigenfunctions are
similar to the functions shown in the figure.
15.u(x,t)=∑∞
n= 1(
ancos(μnt)+bn(sinμnt))
sin(λnx):λn=nπ/a,μn=√
λ^2 n+γ^2 c,an= 2 h(
1 −cos(nπ))
/nπ,bn=0,n= 1 , 2 ,....- Convergence is uniform because
∑
|bn|converges.Section 3.3
- Table showsu(x,t)/h.
t
x 00. 2 a/c 0. 4 a/c 0. 8 a/c 1. 4 a/c
0. 25 a 0.5 0.5 0.2 − 0. 5 − 0. 2
0. 5 a 1.0 0.6 0.2 − 0. 6 − 0. 23.u( 0 , 0. 5 a/c)=0;u( 0. 2 a, 0. 6 a/c)= 0. 2 αa;u( 0. 5 a, 1. 2 a/c)=− 0. 2 αa.
(Hint:G(x)=αx,0<x<a.)5.G(x)={ 0 , 0 <x< 0. 4 a,
5 (x− 0. 4 a), 0. 4 a<x< 0. 6 a,
a, 0. 6 a<x<a.
Notice thatGis a continuous function whose graph is composed of line
segments.