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### Mathematic Methods HW Solutions 36

Course: MHF 2312, Fall 2011
School: UNF
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Word Count: 139

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7 36 10.1 Chapter p(t) = an cos 220nt, a0 = 0 2 2 an = n (sin n + sin 2n ) = n { 3, 0, 0, 0, 3, 0, and repeat} 3 3 1 1 Relative intensities = 1 : 0 : 0 : 0 : 25 : 0 : 49 : 0 : 0 : 0 1 10.2 p(t) = bn sin 262nt, where 1 2 bn = n (1 cos n 3 cos n + 3 cos 2n ) 3 3 2 = n {2, 3, 8, 3, 2, 0, and repeat} 9 4 Relative intensities = 4 : 9 : 64 : 16 : 25 : 0 4 9 10.3 p(t) = bn sin 220nt 2 bn = 5 cos n + 2...

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7 36 10.1 Chapter p(t) = an cos 220nt, a0 = 0 2 2 an = n (sin n + sin 2n ) = n { 3, 0, 0, 0, 3, 0, and repeat} 3 3 1 1 Relative intensities = 1 : 0 : 0 : 0 : 25 : 0 : 49 : 0 : 0 : 0 1 10.2 p(t) = bn sin 262nt, where 1 2 bn = n (1 cos n 3 cos n + 3 cos 2n ) 3 3 2 = n {2, 3, 8, 3, 2, 0, and repeat} 9 4 Relative intensities = 4 : 9 : 64 : 16 : 25 : 0 4 9 10.3 p(t) = bn sin 220nt 2 bn = 5 cos n + 2 cos n ) = n {1, 10, 1, 0, and repeat} 2 1 1 1 1 Relative intensities = 1 : 25 : 9 : 0 : 25 : 25 : 49 : 0 : 81 : 1 9 2 n (3 10.4 V (t) = 200 1+ 2 even n 2 cos 120nt 1 n2 Relative intensities = 0 : 1 : 0 : 10.5 I (t) = 5 1+ 2 even n 1 25 3 : 0 : ( 35 )2 5 2 cos 120nt + sin 120t 2 1n 2 10 Relative intensities = 5 ( )2 : ( 3 )2 : 0 : ( 32 )2 : 0 : ( 72 )2 2 = 6.25 : 1.13 : 0 : 0.045 : 0 : 0.008 10.6 V (t) = 50 400 2 1 odd n 1 cos 120nt n2 Relative intensities = 1 : 0 : 14 3 14 5 :0: 20 (1)n sin 120nt 1 n 1 1 Relative intensities = 1 : 1 : 9 : 16 : 4 10.7 I (t) = 5 20 10.8 I (t) = 2 2 1 odd n 1 25 10 1 cos 120nt n2 1 (1)n sin 120nt n 4 11 4 :: 1+ 2 2 49 9 = 1.4 : 0.25 : 0.12 : 0.06 : 0.04 Relative intensities = 10.9 V (t) = 400 1 odd n 1+ 200 2 1 1 : 16 25 1 sin 120nt n Relative intensities = 1 : 0 : 10.10 V (t) = 75 : 1 odd n 1 9 :0: 1 25 100 1 cos 120nt n2 1 1 sin 120nt n Relative intensities as in problem 10.8 11.5 2 /8 11.8 4 /96 11.6 4 /90 1 2 11.9 16 2 11.7 2 /6 1+ 4 25 2
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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
UNF - MHF - 2312
Chapter 143.13.53.93.123.173.203.236612+i11(a) 5 (1 + 2i)3(a) 0(a) 072i3.2 (2 + i)/33.6 1, 13.10 (2i 1)e2i(b) 1 (8i + 13)3(b) i(b) 17i/43.24 17i/963.3 03.7 (1 i)/83.11 2i3.43.8i/2i/23.18 i 63/3.22 i 3/1083.16 03.19 16
UNF - MHF - 2312
Chapter 14676.286.306.326.346.146.186.226.266.276.316.3511R(3i) = 16 + 24 i6.29R(0) = 1/6!6.31R(2i) = 3ie2 /326.33R(0) = 7, R(1/2) = 76.35 i/46.15 06.1606.19 06.2006.23 sinh 2 6.2433 2 i(1 + cosh 3 + i 3 sinh 3 )316.29
UNF - MHF - 2312
Chapter 146810.4 T = 200 1 arc tan(y/x)10.5 V = 200 1 arc tan(y/x)2210.6 T = 100y/(x + y ); isothermals y/(x2 + y 2 ) = const.;ow lines x/(x2 + y 2 ) = const.10.7 Streamlines xy = const.; = (x2 y 2 )V0 , = 2xyV0 , V = (2ix 2jy )V010.9 Streamlines
UNF - MHF - 2312
Chapter 151.11.31.51.71.91/10, 1/91/3, 5/91/4, 3/4, 1/3, 1/29/26, 1/2, 1/133/10, 1/31.21.41.61.81.103/8, 1/8, 1/41/2, 1/52, 2/13, 7/1327/52, 16/52, 15/529/100, 1/10, 3/100, 1/103/82.122.142.152.17(a) 3/4(b) 1/5(c) 2/3(d) 3/4(e
UNF - MHF - 2312
Chapter 15704.94.124.174.2125/102, 25/77, 49/101, 12/254.11 0.097, 0.37, 0.67; 1354.14 n!/nnMB: 16, FD: 6, BE: 104.18 MB: 125, FD: 10, BE: 35C (n + 2, n)4.22 0.1354.23 0.305.1 = 0, = 35.2 = 7, = 35/65.3 = 2, = 25.4 = 1, = 21/25.5 = 1, =
UNF - MHF - 2312
Chapter 159.3Number of particles:012345Number of intervals: 406 812 812 541 271 1089.4 P0 = 0.018, P1 = 0.073, P4 = 0.1959.5 P0 = 0.37, P1 = 0.37, P2 = 0.18, P3 = 0.069.6 Exactly 5: 64 days. Fewer than 5: 161 days. Exactly 10: 7 days. Moretha
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