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### Mathematic Methods HW Solutions 4

Course: MHF 2312, Fall 2011
School: UNF
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1 15.30 Chapter (a) T = F (5/x + x/40 x3 /16000 ) 1 (b) T = 2 (F/)(1 + 2 /6 + 74 /360 ) 15.31 (a) nite (b) innite 16.1 (c) overhang: 2 3 10 100 books needed: 32 228 2.7 108 4 1086 16.4 C, = 0 16.5 D, an 0 16.6 C, cf. n3/2 1 16.7 D, I = 16.8 D, cf. n 16.9 1 x &lt; 1 16.10 |x| &lt; 4 16.11 |x| 1 16.12 |x| &lt; 5 16.13 5 &lt; x 1 16.14 1 x2 /2 + x3 /2 5x4 /12 16.15 x2 /6 x4 /180...

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1 15.30 Chapter (a) T = F (5/x + x/40 x3 /16000 ) 1 (b) T = 2 (F/)(1 + 2 /6 + 74 /360 ) 15.31 (a) nite (b) innite 16.1 (c) overhang: 2 3 10 100 books needed: 32 228 2.7 108 4 1086 16.4 C, = 0 16.5 D, an 0 16.6 C, cf. n3/2 1 16.7 D, I = 16.8 D, cf. n 16.9 1 x < 1 16.10 |x| < 4 16.11 |x| 1 16.12 |x| < 5 16.13 5 < x 1 16.14 1 x2 /2 + x3 /2 5x4 /12 16.15 x2 /6 x4 /180 x6 /2835 16.16 1 x/2 + 3x2 /8 11x3 /48 + 19x4 /128 16.17 1 + x2 /2 + x4 /4 + 7x6 /48 16.18 x x3 /3 + x5 /5 x7 /7 16.19 (x + ) (x )3 /3! (x )5 /5! 16.20 2 + (x 8)/12 (x 8)2 /(25 32 ) + 5(x 8)3 /(28 34 ) 16.21 e[1 + (x 1) + (x 1)2 /2! + (x 1)3 /3! ] 16.22 arc tan 1 = /4 16.23 1 (sin )/ = 1 16.24 eln 3 1 = 2 16.25 2 16.26 1/3 16.27 2/3 16.28 1 16.29 6! 16.30 (b) For N = 130, 10.5821 < (1.1) < 10.5868 16.31 (a) 10430 terms. For N = 200, 100.5755 < (1.01) < 100.5803 16.31 (b) 2.66 1086 terms. For N = 15, 1.6905 < S < 1.6952 200 86 16.31 (c) ee = 103.138210 terms. For N = 40, 38.4048 < S < 38.4088 4
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UNF - MHF - 2312
Chapter 2xy4.14.24.34.44.54.64.74.84.94.104.114.124.134.144.154.164.174.184.194.201113001322320215104.692.39113124002212 31200131.716.585.15.25.35.45.55.65.75.85.95.105.115.125.13
UNF - MHF - 2312
Chapter 26(2 + 3i)/13; (x yi)/(x2 + y 2 )(5 + 12i)/169; (x2 y 2 2ixy)/(x2 + y 2 )2(1 + i)/6; (x + 1 iy )/[(x + 1)2 + y 2 ](1 + 2i)/10; [x i(y 1)]/[x2 + (y 1)2 ](6 3i)/5; (1 x2 y 2 + 2yi)/[(1 x)2 + y 2 ](5 12i)/13; (x2 y 2 + 2ixy )/(x2 + y 2 )15.2
UNF - MHF - 2312
Chapter 29.19.49.79.109.139.169.199.229.259.309.339.3610.110.310.510.710.910.1010.1110.1310.1510.1710.1810.2010.2210.2310.2410.2510.2610.2811.311.7(1 i)/ 29.2 i9.3 9ie(1 + i 3)/29.5 19.6 13 e29.8 3 + i9.9 2i29.1
UNF - MHF - 2312
Chapter 214.114.314.514.714.914.1114.1214.1314.1414.1514.1614.1714.1814.2014.2215.115.215.315.415.515.615.715.815.915.1015.1115.1215.1315.1415.1515.1616.216.316.416.516.616.716.816.916.1016.121 + i14.2 i/2 or 3i/
UNF - MHF - 2312
Chapter 2917.1 1 17.2 ( 3 + i)/217.3 r = 2, = 45 + 72 n : 1 + i, 0.642 + 1.260i, 1.397 0.221i,0.221 1.397i, 1.260 0.642i17.4 i cosh 1 = 1.54i17.5 i22217.6 e = 5.17 105 or e e2n17.7 e/2 = 4.81 or e/2 e2n17.8 117.9 /2 2n217.11 i17.10 3 17.
UNF - MHF - 2312
Chapter 32.310013, x = 3, y = 5)52.410011/2 1/2,012.5101/2 1/2 0,0012.62.72.82.92.102.112.122.132.14x = (z + 1)/2, y = 1no solution1001, x = 1, z = y0 1 1 01 0 40 13, x = 4, y = 30001 1 0 110017, x = y 11, z
UNF - MHF - 2312
Chapter 32.15 R = 22.17 R = 2112.16 R = 32.18 R = 33.13.53.163.173.18113.2 7213.3 13.4 21405443.6 43.11 03.12 16A = (K + ik )/(K ik ), |A| = 1x = (x + vt ), t = (t + vx /c2 )D = 3b(a + b)(a2 + ab + b2 ), z = 1(Also x = a + 2b, y = a
UNF - MHF - 2312
Chapter 35.335.355.375.395.405.415.435.456.16.26.36.46.5125.3411/10 43/1555.36 3Intersect at (1, 3, 4)5.38 arc cos 21/22 = 12.3t1 = 1, t2 = 2, intersect at (3, 2, 0), cos = 5/ 60, = 49.8t1= = 1, t2 = 1, intersect at (4, 1, 1), cos
UNF - MHF - 2312
Chapter 3136.85x2 + 3y 2 = 306.9AB =00006.10 AC = A D =BA =11 1233 365/3 3124516.15 2 22232216.17 A1 = 4 46823B1 AB = 22116.19 A1 =37146.20 A1 =57116.21 A1 = 2534176.22 A1 =1216.13224411 221310
UNF - MHF - 2312
Chapter 31410000 , S = 0 0 1 ; R is a 90 rotation about0101a 90 rotation about the x axis.0 100010 1 ;From problem 30, RS = 1 0 0 , SR = 0100010RS is a 120 rotation about i + j + k; SR is a 120 rotation about i j + k.180 rotation about
UNF - MHF - 2312
Chapter 31510.3 (a) Label the vectors A, B, C, D. Then cos(A, B) =cos(A, C) =23 , cos(B, C)3 , cos(A, D) =2317, cos(C, D) = 621 .69023=1 ,152,3 15cos(B, D) =(b) (1, 0, 0, 5, 0, 1) and (0, 0, 1, 0, 3, 0)10.4 (a) e1 = (0, 1, 0, 0), e2 =
UNF - MHF - 2312
Chapter 31 1 20, C= 11521 1 10D=, C= 21211 1 10D=, C= 11211210, C= D=25 1 21 1i = 1, 3; U = 2i1111i = 1, 4; U = 13 1 i12 i = 2, 3; U = 5 i 2153 4i = 3, 7; U = 55 2 3 4i1i2 11U = 1 i 2 12202Reection t
UNF - MHF - 2312
Chapter 31713.5 The 4s group13.6 The cyclic group13.7 The 4s group13.10 If R = 90 rotation, P = reection through the y axis, and Q = PR, then the8 matrices of the symmetry group of the square are:100 110I=,R=, R2 == I,01100 1011 001R
UNF - MHF - 2312
Chapter 318101 i15.7 AT =A=101iA1 =1 i0 iBT AT = (AB)T2 63i 5i221BT AC = 1 3i1 5i 1AB =20020 iC 1 A =BT C = 311 15 1AT BT , BAT , ABC, ABT C, B1 C, and CBT are meaningless.1 i13i0 6i10 A1 =1 i 215.8 A = 0 33i2i0
UNF - MHF - 2312
Chapter 4221.11.21.31.41.51.71.101.131.161.191.221.71.101.131.141.171.191.211.23u/x = 2xy 2 /(x2 + y 2 ) , u/y = 2x2 y/(x2 + y 2 )s/t = utu1 , s/u = tu ln tz/u = u/(u2 + v 2 + w2 )At (0, 0), both = 0; at (2/3, 2/3), both = 4At (0
UNF - MHF - 2312
Chapter 4206.46.56.76.96.11y = y (x 1)/[x(y 1)], y = (y x)(y + x 2)y/[x2 (y 1)3 ]2x + 11y 24 = 06.6 1800/113y = 1, x y 4 = 06.8 8/3y = x 4 2, y = 0, x = 06.10 x + y = 0y =47.17.27.37.47.57.67.77.87.107.117.127.13dx/dy = z y + ta
UNF - MHF - 2312
Chapter 4219.69.89.109.12V = 1/3(8/13, /13)12d = 5/ 2Let legs of right triangle be a andthen a = b, h = 2 2 a.9.7 V = d3 / abc)(279.9 A = ab 3/439.11 d = 6/2b, height of prism = h;10.110.310.510.710.9d=12, 14d 1=12 11max T = 4
UNF - MHF - 2312
Chapter 413.8 1 ft 4 inches=13.9 dz/dt = 1 + t(2 x y )/z , z = 013.10 (x ln x y 2 /x)xy where x = r cos , y = r sin b2 x d2 yb4dy= 2 ,= 2 313.11dxa y dx2ay13.12 1313.13 113.14 (w/x)y = (f /x)s, t + 2(f /s)x, t + 2(f /t)x,13.15 (w/x)y = f
UNF - MHF - 2312
Chapter 52.12.52.92.132.172.212.252.292.332.372.412.452.493.23.33.43.53.63.73.83.93.103.113.143.153.173.193.213.223.233.243.253.263e2541267/4(ln 2)/2131/63/2216/37/6546k/151/32.2182.342.48/32.62.35
UNF - MHF - 2312
Chapter 5(b) x = y = 4a/(3 )(c) I = M a2 /4(e) x = y = 2a/4.2 (c) y = 4a/(3 )(d) Ix = M a2 /4, Iy = 5M a2 /4, Iz = 3M a2 /2(e) y = 2a/(f) x = 6a/5, Ix 48M a2 /175, Iy = 288M a2 /175, Iz = 48M a2 /25=2(g) A = ( 3 1 3)a2214.3 (a), (b), or (c) 2
UNF - MHF - 2312
Chapter 56.176.196.206.216.226.236.246.256.266.27M a2 / 66.18 (0, 0, 5h/6)2Ix = Iy 20M h /21, Iz = 10M h2 /21, Im = 65M h2 /252=(b) 3/2(a) (5 5 1)/6Gh(2 2)Ix = M b2 /4, Iy = M a2 /4, Iz = M (a2 + b2 )/4(a) (0, 0, 2c/3)(b) (0, 0, 5c/7)
UNF - MHF - 2312
Chapter 63.13.23.33.43.53.63.73.83.93.123.153.163.173.193.20(A B)C = 6C = 6(j + k), A(B C) = 2A = 2(2, 1, 1),(A B) C = A (B C) = 8, (A B) C = 4(j k),A (B C) = 4(i + 2k)B C = 16(A + B) C = 5B A = i + 7j + 3k, |B A= 59, (B A) C/|C| = 8/
UNF - MHF - 2312
Chapter 6276.116.126.136.146.15 = 2xi 2y j, E = 2xi + 2y j(a) 2 5 , 2i + j(b) 3i + 2j(a) i + j, | | = e (b) 1/2(b) Down, at the rate 11 2(a) 4 2 up,(b) 0, around(c) 4/ 10, down (d) 8/5, up6.17 er6.18 i7.17.37.57.67.77.87.107.127.
UNF - MHF - 2312
Chapter 611.111.411.711.1011.133a2120602811.211.511.811.1111.142ab236024811.311.611.911.1211.1504532/318 2 2In the answers for Problems 18 to 22, u is arbitrary.11.18 A = (xz yz 2 y 2 /2)i + (x2 /2 x2 z + yz 2 /2 yz )j +1
UNF - MHF - 2312
Chapter 7amplitudeperiodfrequencyvelocityamplitude2.132/55/(2 )152.22/22/82.31/221/2/22.4521/(2 )51/33/2.52.6s = sin 6ts = 6 cos8sin 2t6 cos8= 5.541/612 cos8= 11.12.7521/(2 )52.8241/(4 )12.9221/222
UNF - MHF - 2312
Chapter 7305.1 to 5.11The answers for Problems 5.1 to 5.11 are thesine-cosine series in Problems 7.1 to 7.11.x2/20/226.11/21/211/201/21/26.21/2001/21/201/26.301/2001/21/206.410110016.51/21/201/201/21/26.6
UNF - MHF - 2312
Chapter 77.5312an = n sin n2bn =cn =a 0 / 2 = c0 = 0n1n 2 cos 2 1in/22in 2ec n = cn41 cos n = n cfw_0, 1, 0, 0, and repeat 1 ein =1n cfw_1, 2i, 1, 0,and repeat , n &gt; 01f (x) = eix + eix 7.67.72i2ix e2ix 1 (e3ix + e3ix )2e32i
UNF - MHF - 2312
Chapter 78.3321an = n sin n , a0 /2 = c0 = 12411bn = n cos n cos n = n cfw_1, 2, 1, 0 , and repeat2icn = 2n (ein ein/2 )1= 2n cfw_(1 + i), 2i, 1 i, 0, and repeat, n &gt; 0; cn = cnf (x) =1412+(1 + i)eix/l (1 i)eix/l +2i 2ix/l2 (e e2ix
UNF - MHF - 2312
Chapter 7338.11 (a) f (x) =(b) f (x) =22(1)n inx+2e=+43n23n=04 2+231i+n2n(1)ncos nxn214 2+43einx =1n=08.12 (a) f (x) ==(1)n (1 + in) inxe1 + n2sinh sinh 2 sinh +e2 1(b) f (x) =2=(b) f (x) =2in=0n=088.14
UNF - MHF - 2312
Chapter 7348.20 f (x) =2+3an cos12nx+310 , n = 3 kan =9, otherwise8 n2 29.19.29.39.59.7(a) cos nx + i sin nxx(a) 1 ln |1 x2 | + 1 ln | 1x |221+(a) (x4 1) + (x5 + x3 )14sin nxf (x) =1nan =f (x)9.8odd n2sin n , a0 /2 =
UNF - MHF - 2312
Chapter 7359.18 continuedFunction of period 3:31an = n sin 2n = 2n cfw_1, 1, 0, and repeat, a0 /2 = 1/3313bn = n (1 cos 2n ) = 2n cfw_1, 1, 0, and repeat33x1+ 2 (cos 2x 1 cos 4x + 1 cos 8x 1 cos 10333234351x+ 23 (sin 2x + 1 sin 4x
UNF - MHF - 2312
Chapter 73610.1 p(t) =an cos 220nt, a0 = 022an = n (sin n + sin 2n ) = n cfw_ 3, 0, 0, 0, 3, 0, and repeat3311Relative intensities = 1 : 0 : 0 : 0 : 25 : 0 : 49 : 0 : 0 : 0110.2 p(t) =bn sin 262nt, where12bn = n (1 cos n 3 cos n + 3 cos 2
UNF - MHF - 2312
Chapter 712.412.512.612.712.812.912.1012.1112.1212.1312.1412.1512.1612.1712.1812.1912.2012.2112.2512.2712.2821 cos sin x d01 cos ixf (x) =e disin sin(/2) ixe df (x) =1 ei ixf (x) =e d2isin cos ixe df (x) =i2cos + s
UNF - MHF - 2312
Chapter 73822(b) fs (x) =112.30 (a) fc (x) =1(b) fs (x) =sin 3 2 sin 2cos x d02 cos 2 cos 3 1sin x d01 cos 2cos x d202 sin 2sin x d2012.29 (a) fc (x) =13.2 f (x) =i21 2inxenn =013.4 (c) q (t) = CV 1 2(1 e1/2 )13.6 f (t) =(
UNF - MHF - 2312
Chapter 739cos 2 12 cos 2 1, f (x) =sin x d,i08 cos sin2 (/2)cos x d, /813.16 f (x) =0213.15 g () =13.190|f (x)|2 dx =0|gc ()|2 d =2 sin2 a13.20 g () =, a3 /3 20|gs ()|2 d13.23 2 /8/4
UNF - MHF - 2312
Chapter 81.41.51.61.7x = k 1 gt + k 2 g (ekt 1)x = A 2 sin t + v0 t + x0(a) 15 months (b) t = 30(1 21/3 ) = 6.19 monthsx = (c/F )[(m2 c2 + F 2 t2 )1/2 mc]2.12.32.52.72.92.112.132.152.172.19y = mx, m = 3/22.22.4ln y = A(csc x cot x),
UNF - MHF - 2312
Chapter 84113.13 x = 2 ey + Cey3.14 x = y 2/3 + Cy 1/313.15 S = 2 107 (1 + 3t/104) + (1 + 3t/104 )1/3 , where S = numberof pounds of salt, and t is in hours.3.16 I = AeRt/L + V0 (R2 + 2 L2 )1 (R cos t + L sin t)3.17 I = Aet/(RC ) V0 C (sin t RC c
UNF - MHF - 2312
Chapter 842y = (Ax + B ) sin x + (Cx + D) cos x + (Ex + F )ex + (Gx + H )ex = 0 cos t, = g /lT = 2 R/g 85 min.= = 1/ LCoverdamped: R2 C &gt; 4L; critically damped: R2 C = 4L;underdamped: R2 C &lt; 4L.2t4 22t16y+y = 0, y = e8t/15 A sin+ B cos5.40
UNF - MHF - 2312
Chapter 87.107.117.127.137.167.177.187.197.207.217.227.237.257.288.88.108.128.178.228.258.2743x = 1 + t2x = (1 3t)1/3xt = 1 2 (1 u4 )1/2 duut = ( 2 )1 (cos )1/2 d(a) y = Ax + Bx3(b) y = Ax2 + Bx2(c) y = (A + B ln x)/x3(d) y
UNF - MHF - 2312
Chapter 810.310.510.710.810.910.1110.1310.154412 t sinh t10.4b(b a)tebt + a[ebt eat ]10.6(b a)2ata cosh bt b sinh bt aea2 b 2eatebtect++(b a)(c a) (a b)(c b) (a c)(b c)(2t2 2t + 1 e2t )/410.10 (1 cos at 1 at sin at)/a421cos bt
UNF - MHF - 2312
Chapter 8x 2 sin x,x &lt; /412.13 y = x 2 cos x, x &gt; /4212.15 y = x sinh x cosh x ln cosh x12.16 y = x ln x x x(ln x)2 /22112.17 y = 4 sin x12.18 y = x2 /2 + x4 /61linear 1st ordery = 3 x2 + Cx2(ln y ) (ln x)2 = Cseparabley = A + Bex sin(x
UNF - MHF - 2312
Chapter 92.12.32.52.62.72.9(y b)2 = 4a2 (x a2 )2.2 x2 + (y b)2 = a2ax = sinh(ay + b)2.4 ax = cosh(ay + b)y = aex + bex or y = A cosh(x + B ), etc.x + a = 4 (y 1/2 2b)(b + y 1/2 )1/23ex cos(y + b) = C2.8 K 2 x2 (y b)2 = K 42x = ay + b2.10
UNF - MHF - 2312
Chapter 95.55.65.85.95.105.115.125.135.145.155.165.175.185.195.205.215.225.235.245.2511L = 2 mx2 2 kx2mx + kx = 01L = 2 m(r2 2 + r2 sin2 2 ) mgr cos a sin cos 2 g sin = 0a(d/dt)(sin2 ) = 012L = 2 m(2r + r2 2 ) mgr2 + g =
UNF - MHF - 2312
Chapter 96.16.4catenarycatenary8.2I=8.3I=8.4I=486.26.5x2 y21+yy dy2x2+1circlecircle6.36.6circular cylindercircledx, x2 (2y + y 3 ) = K (1 + y 2 )3/222, x y 2 = C 2 (1 + x )3r2 + r4 2 dr ,drd= Kr r4 K 2y = aebx(x a)2 + (y
UNF - MHF - 2312
Chapter 104.44.54.64.7 1 0221 0 Principal moments:I=( 1, , + 1); principal151500axes along the vectors: (1, 1, 0), (0, 0, 1), (1, 1, 0).4004 2 Principal moments: (2, 4, 6); principal axes along theI = 00 24vectors: (0, 1, 1), (1, 0
UNF - MHF - 2312
Chapter 108.58.68.78.88.98.118.128.138.1450V = er cos e sin e r sin hu = hv = (u2 + v 2 )1/2 , hz = 1ds = (u2 + v 2 )1/2 (eu du + ev dv ) + ez dzdV = (u2 + v 2 ) du dv dzau = iu + jv = (u2 + v 2 )1/2 euav = iv + ju = (u2 + v 2 )1/2 evaz =
UNF - MHF - 2312
Chapter 109.351See 8.21 UU1 U+ e+ err r sin 111 VV = 2r2 Vr +(sin V ) +r rr sin r sin U1U12U12r2+2sin +2 2U= 2r rrr sin r sin 21VV =er(sin V ) r sin 1Vr1 Vr sin (rV ) e +(rV ) +er sin rr r9.6 See 8.11
UNF - MHF - 2312
Chapter 109.189.199.209.2152 r 1 e , r 1 e r , 32e , er cos e sin , 3r 1 , r 3 , 02r1 , 6, 2r4 , k2 eikr cos 11.4 Vector11.5 ds2 = du2 + h2 dv 2 , hu = 1, hv = u(2v v 2 )1/2 ,vdA = u(2v v 2 )1/2 du dv, ds = eu du + hv ev dv ,eu = i(1 v ) + j
UNF - MHF - 2312
Chapter 113.23.53.83.113.143/232/35(5/3)1(4/3)3.33.63.93.123.159/1072(5/4)(2/3)/3(2/3)/417.12 B (5/2, 1/2) = 3/1617.33 B (1/3, 1/2)7.5 B (3, 3) = 1/3017.72 B (1/4, 1/2)7.10 4 B (1/3, 4/3)37.12 (8/3)B (5/3, 1/3) = 32 2 3/27
UNF - MHF - 2312
Chapter 115412.15 2 E 23 E12.14 12E 35 15.86=12.16 2 2 E 1/ 2 3.820=a12.23 T = 8 5g K 1/ 5 ; for small vibrations, T 2 2a=3g13.713.913.1113.1313.1513.1713.1913.2113.2213.24(4) = 3!3/22.4222E135 F arc sin 4 , 4/5 = 0.1834 sn u d
UNF - MHF - 2312
Chapter 121.11.31.51.71.9yyyyy2.1= a1 xex= a1 x= a0 cosh x + a1 sinh x= Ax + Bx3= a0 (1 x2 ) + a1 x3= a0 e x= a0 cos 2x + a1 sin 2x= (A + Bx)ex= a0 (1 + x) + a2 x22= (A + Bx)ex1.21.41.61.81.10yyyyySee Problem 5.32.4Q0 =
UNF - MHF - 2312
Chapter 125610.5 sin (35 cos3 15 cos )/210.4 sin 10.6 15 sin2 cos 11.111.311.511.611.711.811.911.10y = b0 cos x/x211.2 y = Ax3 + Bx332y = Ax + Bx11.4 y = Ax2 + Bx31/21/2y = A cos(2x ) + B sin(2x )y = Aex + Bx2/3 [1 3x/5 + (3x)2 /(5 8
UNF - MHF - 2312
Chapter 1257= Ax + B x sinh1 x x2 + 1= A(1 + x) + Bxe1/x22= A(1 x ) + B (1 + x )exx= Ax Be= A(x 1) + B [(x 1) ln x 4]= A x + B [ x ln x + x]xx= A 1x + B [ 1x ln x + 1+x ]2= A(x2 + 2x) + B [(x2 + 2x) ln x + 1 + 5x x3 /6 + x4 /72 + ]= Ax2 +
UNF - MHF - 2312
Chapter 132.1T=2.2T=2.32.42.7T=nx(1)n+1 ny/10esinn101200 41odd nn22even n22n=2+4knx 1 ny/20sinen20neny sin nx1x 1 3y/30120 y/303x1 5y/305xesinesin+esin2309302530n4sinh n(1 y ) sin nxT= 2 (n2 1) sinh
UNF - MHF - 2312
Chapter 13592.14 For f (x) = x 5: T = 4021odd nnx ny/101cose2n10For f (x) = x: Add 5 to the answer just given.2.15 For f (x) = 100, T = 100 10y/3401For f (x) = x, T = (30 y) 263.23.33.43.53.63.7u=4001odd nu = 100 u=40 1od
UNF - MHF - 2312
Chapter 134.64.74.8y=y=y=604hl2 v9hl3 v4l2 v4.91odd n1nnwnxnvt1sinsinsinsin2n2lllnnxnvt1sinsinsinn33ll1sin(n/2)xvt2x2vtnxnvtsinsin+sinsinsinsin3ll16lln(n2 4)lln=31. n = 1, = v/(2l)2. n
UNF - MHF - 2312
Chapter 136116, n odd, the six solutions on (0, ) aren (4 n2 )1. T = bn eny sin nxbnsinh n(H y ) sin nx2. T =sinh nH23. u =bn e(n) t sin nxh 2 n24. =bn sin nx eiEn t/h , En =2m5. y = bn sin nx cos nvtbn6. y =sin nx sin nvtnv12(1)n+1
UNF - MHF - 2312
Chapter 135.14 u =6250 ln r200+ln 2o dd nln r5.15 u = 50 1 ln 2r n r nsin nn(2n 2n )200o dd nn (2n1 2 n )r2nr2nsin n6.2The rst six frequencies are 10 , 11 = 1.59310 , 12 = 2.1351020 = 2.29510 , 13 = 2.65210 , 21 = 2.91710.6.4
UNF - MHF - 2312
Chapter 136322227.21 n (x) = e (x +y +z )/2 Hnx (x)Hny (y )Hnz (z ), = m/ ,h1113En = (nx + 2 + ny + 2 + nz + 2 ) = (n + 2 ) .hh(n + 2)(n + 1), n = 0 to .Degree of degeneracy of En is C (n + 2, n) =2M e4+12r7.22 (r, , ) = R(r)Ylm (, )
UNF - MHF - 2312
Chapter 1310.12 u =40064o dd n1rna2nsin 2n10.14 Same as 9.1210.15 u =1rn 104006nsin 6n =2(10r)6 sin 6200arc tan1012 r12o dd n10.16 v 5/(2 )10.17 mn , n = 0; the lowest frequencies are:11 = 1.5910 , 12 = 2.1410 , 13 = 2.6510, 21 = 2.
UNF - MHF - 2312
Chapter 141.11.31.51.71.81.91.101.111.121.131.151.171.181.191.201.21u = x3 3xy2 , v = 3x2 y y 31.2 u = x, v = yu = x, v = y1.4 u = (x2 + y 2 )1/2 , v = 0u = x, v = 01.6 u = ex cos y , v = ex sin yu = cos y cosh x, v = sin y sinh xu