1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

448 Answers to Odd-Numbered Exercises Section 1.3 a. sectionally smooth; b, c, d, e are not; b: vertical tangent at 0; c: verti ...

Chapter 1 449 Section 1.6 π^1 ∫π −π ( ln ∣∣ ∣∣2cos ( x 2 )∣∣ ∣∣ ) 2 dx= ∑∞ n= 1 1 n^2 = π^2 6. a. Coefficients tend to zero. ...

450 Answers to Odd-Numbered Exercises Figure 1 Graph for Exercise 3, Section 1.7. is a product of continuous functions and is th ...

Chapter 1 451 b.sin(x) x = ∫∞ 0 A(λ)cos(λx)dλ,whereA(λ)= { 1 , 0 <x<1, 0 , 1 <x. a.A(λ)≡0,B(λ)=2sin(λπ ) π( 1 −λ^2 ) ...

452 Answers to Odd-Numbered Exercises Chapter 1 Miscellaneous Exercises 1.f(x)= ∑∞ n= 1 bnsin(nx), bn= { 0 , neven, 4sin(nα) παn ...

Chapter 1 453 13.f(x)= ∑∞ n= 1 bnsin(nπx),bn= 2 ( 1 +cos(nπ) ) /nπ. 15.f(x)= ∑∞ n= 1 bnsin(nx), b 2 =^1 2 ,otherbn=4sin(nπ/^2 ) ...

454 Answers to Odd-Numbered Exercises 35.a 0 = a 6 ,an= 2 a n^2 π^2 ( cos ( 2 nπ 3 ) −cos (nπ 3 )) . 37.a 0 =^58 ,an=n (^22) π 2 ...

Chapter 2 455 Chapter 2 Section 2.1 One possibility:u(x,t)is the temperature in a rod of lengthawhose lat- eral surface is insu ...

456 Answers to Odd-Numbered Exercises Section 2.3 1.w(x,t)=−π^2 (T 0 +T 1 )sin ( πx a ) exp ( −π (^2) kt a^2 ) −^2 π (T 0 −T 1 2 ...

Chapter 2 457 c.A=(S 1 −S 0 )/a,B=S 0 .IfS 0 =S 1 ,then∂∂ut=kAfor allt. 7.φ′′+λ^2 φ=0, 0<x<a, φ( 0 )=0,φ(a)=0. Solution:φ ...

458 Answers to Odd-Numbered Exercises In the integral forbn, break the interval of integration ata; in the second integral, make ...

Chapter 2 459 Section 2.8 1.x= ∑∞ n= 1 cnφn,1<x<b;cn= 2 nπ^1 −bcos(nπ) n^2 φ^2 +ln^2 (b) . 3. 1= ∑∞ n= 1 cnφn,0<x<a; ...

460 Answers to Odd-Numbered Exercises w( 0 ,t)=0, 0 <t, w(x, 0 )=−C 0 e−ax,0<x; c. w(x,t)=e−a^2 Dt ∫∞ 0 B(λ)sin(λx)e−λ^2 D ...

Chapter 2 461 u(x,t)=T 0 + ∑∞ 1 bnsin(λnx)e−λ^2 nkt, bn=^2 a ∫a 0 (T 1 −T 0 )sin ( nπx a ) dx. SS:v(x)=T 0 + r 2 x(x−a),0<x ...

462 Answers to Odd-Numbered Exercises u(x,t)=√T^0 4 πkt ∫a 0 exp ( −(x ′−x) 2 4 kt ) dx′ =T^0 2 [ erf ( a√−x 4 kt ) +erf ( √x 4 ...

Chapter 3 463 c.w(x,t)= ∑ cnφn(x)e−λ^2 nkT, φn(x)=e−μx/^2 sin(nπx/L), λ^2 n= (nπ L ) 2 +μ 2 4 ; d.λ^2 n=( 7. 30 n^2 + 0. 0133 )× ...

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(μ ...

Chapter 3 465 Figure 2 Solution for Exercise 7, Section 3.3. Figure 3 Solution for Exercise 9, Section 3.3. See Fig. 2. See Fig ...

466 Answers to Odd-Numbered Exercises and similarly ∂^2 u ∂t^2 =c 2 (∂ (^2) v ∂w^2 −^2 ∂^2 v ∂z∂w+ ∂^2 v ∂z^2 ) . (We have assum ...

Chapter 3 467 Figure 4 Solution for Exercise 3, Section 3.6. Givenα,pcan be adjusted so thatmis an integer whenevernis an intege ...