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468 Answers to Odd-Numbered Exercises Figure 5 Solution for Exercise 5, Section 3.6. Figure 6 Solution of Miscellaneous Exercise ...

Chapter 3 469 Figure 7 Solution of Miscellaneous Exercise 5, Chapter 3. Figure 8 Solution of Miscellaneous Exercise 7, Chapter 3 ...

470 Answers to Odd-Numbered Exercises Figure 9 Solution of Miscellaneous Exercise 7, Chapter 3. Figure 10 Solution for Miscellan ...

Chapter 4 471 an=^8 aU^0 (−^1 ) n+ 1 π^2 ( 2 n− 1 )^2 ,bn=0. 23. Y′′ Y = 2 V k ψ′ ψ.Thefunctionφ(x−Vt)cancels from both sides. 2 ...

472 Answers to Odd-Numbered Exercises Membrane is attached to a frame that is flat on the left and right but has the shape of th ...

Chapter 4 473 a. See Eq. (11).an=0,cn= 200 ( 1 −cos(nπ ))/nπ; b.u(x,y)=u 1 (x,y)+u 2 (x,y),u 1 (x,y)is the solution to Part a, ...

474 Answers to Odd-Numbered Exercises Check zero boundary conditions by substituting. Atx=a,find Ancosh(μna)=^2 b ∫b 0 Sycos(μ ...

Chapter 4 475 b.u(x,y)= 2 π ∫∞ 0 λ 1 +λ^2 sin(λx) sinh(λ(b−y)) sinh(λb) dλ. 11.u(x,y)= ∫∞ 0 2 π( 1 +λ^2 ) sinh(λx) sinh(λa)cos(λ ...

476 Answers to Odd-Numbered Exercises c.u(x,y)= ∑∞ 1 ansin(nπx)exp(−n^2 π^2 y), an= 2 ∫ 1 0 f(x)sin(nπx)dx. 7.X′′/X=−λ^2 ,T′′/T= ...

Chapter 4 477 21.u(r,θ)= ∑∞ 1 bn (r c )n/ 2 sin(nθ/ 2 ),bn=^1 π ∫ 2 π 0 f(θ )sin(nθ/ 2 )dθ. 23.u(x,y)= ∑ cnsinh(λny)sin(λnx),λn= ...

478 Answers to Odd-Numbered Exercises a.u=−r 2 4 +c 1 ln(r)+c 2 ; b.u=−(ln(r)) 2 2 +c 1 ln(r)+c 2. 39.V(x,y)∼=a 0 =^1 a ∫a 0 f ...

Chapter 5 479 Section 5.3 Ifa=b, the lowest eigenvalues are those with indices(m,n),inthisor- der:( 1 , 1 );( 1 , 2 )=( 2 , 1 ) ...

480 Answers to Odd-Numbered Exercises Taking the hint and using the fact that∇^2 φ=−λ^2 φ, the left-hand side becomes ( λ^2 k−λ ...

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 ...

482 Answers to Odd-Numbered Exercises (A superscriptkin parentheses meanskth derivative.) The right-hand side looks like the bin ...

Chapter 5 483 The first three nonzero terms are, fora=b, those with(m,n)= ( 1 , 1 ), ( 1 , 3 )=( 3 , 1 ), ( 3 , 3 ). All terms w ...

484 Answers to Odd-Numbered Exercises ony= √ 3 ( 1 −x),φ=sin( 4 nπx)+sin( 2 nπ( 1 − 2 x))−sin 2nπ. For a sextant,φn=J 3 n(λr)si ...

Chapter 6 485 Section 6.2 a.e^2 t;b.e−^2 t;c.^3 e −t−e− 3 t 2 ;d. sin( 3 t) 3. a.^1 −e −at a ;b.t−sin(t);c. sin(t)−^12 sin( ...

486 Answers to Odd-Numbered Exercises a. ω ω^2 −π^2 ( 1 πsin(πt)− 1 ωsin(ωt) ) sin(πx); b. 1 2 π^2 ( sin(πt)−πtcos(πt) ) sin(π ...

Chapter 7 487 17.f(t)= ∑∞ −∞ 1 2 a G (inπ a ) einπt/a. 19.F(s)must be of the formF(s)=G(s)/H(s),whereG(s)is never infinite. The ...