Chapter 5 481
5.φ(a,θ)=0andφ(r,θ)=φ(r,θ+ 2 π).- SetJm(λmnr)=φn.Then(rφ′n)′=−λmnrφnand(rφ′q)′=−λmqrφqare the
equations satisfied by the functions in the integrand. Follow the proof in
Section 2.7. - Generally, radii forφ 0 narer=λ 0 m/λ 0 nform= 1 , 2 ,...,n.Forn=2,
r= 2. 405 / 5. 520 = 0 .436 and 1.
Section 5.8
1.φ(x)=xα[AJp(λx)+BYp(λx)], whereα=( 1 −n)/2,p=|α|.- Forλ^2 =0,Z=A+Bz.
5.φ(ρ+ct)=F ̄o(ρ+ct)+G ̄e(ρ+ct),
ψ(ρ−ct)=F ̄o(ρ−ct)−G ̄e(ρ−ct),
whereF ̄o(x)is the odd periodic extension with period 2aofxf(x)/2and
G ̄e(x)is the even periodic extension with period 2aof∫(x/ 2 c)g(x)dx. - The weight function isρ^2 and the interval is 0 toa.
9.v(x)=(b−x)(x−a)/(a+b)x^2. - No. The idea is to find a solution of the partial differential equation that
depends on only one variable. That is impossible iffdepends on bothx
andy.
13.an=−∫b
av(x)Xn(x)x(^3) dx
∫b
aX^2 n(x)x^3 dx
.
Section 5.9
- [k(k+ 1 )−μ^2 ]ak+ 1 −[k(k− 1 )−μ^2 ]ak− 1 =0, valid fork= 1 , 2 ,....
3.P 5 =1
8
(
63 x^5 − 70 x^3 + 15 x)
.
5.y=Aln( 1 +x
1 −x)
.
- Differentiate Eq. (9) and add to itntimes Eq. (8).
- Leibniz’s rule states that
(uv)(k)=∑k
r= 0(
k
r)
u(k−r)v(r).