454=&(1+r)-x—Aj(1+r)-x=[—(I+r)-x](l+r)-x=4(1+ᔉ-x[l+(l+r)]4—A2(1+r)-x={4(l+r)2-x[l+(l+r)]}(l+r)-x=4(1+ᔆ>&%[1+(1+r)+(1+r)2-(᧕*+,-./0ᵨᦪ12345/6789AᨵA“=4(l+r)-x[l+(l+r)+(l+r)2+_+(i+r)T-ᵫᦪᑖ Ḅ (1+ᔆ=y)/-l=()K-l)(l+y+/+...+/-1),H>l,y>l7LᨵMM,1ஹ(1+r)&1ஹ(1+rf-lA,=4(1+ᔆ)-x--=4(1+ry-x^-(1+r)-1rᵫ7AN—°,ᡠU(l+r)N_]V4“ᦻXY”ᵯ\]ᐺ_Ḅ`aLᔲcd.ef,ᦪg/hijMklmnL/ofLepqḄ/rstᑮᙠᦟw⌕e⌕(ᐕzeᐕz){ᵨ|fᘤ |f~tkḄᦪeḄ⚪(q▭LeḄ⚪).ᡃ/ᐕz/ὃeᐕz.
46ᑮhi/gẆ/neLᜩᙠ⌕Ḅ⚪.e/ᡃ¡¢t£ᐵ¥720095ᨴ18ªᵫWolframResearch(´µ¶·¸Ẇ) ¹c º,(»¼)Ḅ&½7Mathematicaᦪ ANewKindofScience(&ÆÇÈ/ÉÊ1280⚓/ÌᑏNKS)ÎWolfram|AlphaḄÇḄ|fÑÒ(ÓÔ)ÕÖ(Computationalknowledgeengine)UtÛÜ6ÈẆ ᦟÝÞßḄàá.WolframiAlphaḄὅStephenWolfram(1959,8,29-,1979ᙠäåᳮç▾(CIT)êᳮᱥᳮìíî/1988ïº,ðñᜧḄ|f~Mathematica),ïᨬôõᦻ⊤“(Wolfram|AlphaḄ)ᵨᡝᡠ⌕ḄrLᵨ÷ᯠḄùú⚪/ûÓÔÕÖᑣýþdÿ .ᡃᐶᙢᔠᵨឋḄ(algorithmsandheuristics),-.(linguisticdiscovery),ᡃ23456789:⌕Ḅᳮ=>ẚ@4A▭CᐸEF.ᡃ2GᨬIJᡂ8LMNOwww.wolframalpha.comSLMNTἠVᓫXᐭZ⚪ᡃ2\3]^ᐭᑮ8L`ᜧḄbcSLbcdeᨵᐸhᜧijkḄᦪmn.”ᐵqrGsᦪtᦟvwxḄyz{3]}ᵫJeffreyR.YoungᑏḄ⊤ᙠ20096ᨴ12ChronicleofHigherEducation(ᦟv)CḄᦻ"ACalculatingWebSiteCouldIgniteaNewCampus'MathWar*(¡MN34¢£¤¥8¦Ḅ‘ᦪtᡊ©‘)”.ᵨMathematicaᦪtª«ḄXᐭ,X¬XᐭOClear[r,n,N,x\x[rnA)=(1+r)n-14=200000;n=N=240;r=0.005325;x[r,N,4]ᭊ¬O1478.22±Ḅ²ᵨ3³ὃ[1]:¸ᜧtxᦪt¹º»¼½¾ᦟᩞ(À)ÁÂ3ÄᐸÅÆÇÈᓭᦟv¬᱐Ë2008.ºÌḄÍJOÎ]p.33,(3.1-4)(3.1-9),4LÍkÐÑ⍝ÓÔ3L\3]Õ¬Ö8L.Y(3.1-4)
47A/(l+r)"(l+r)H-l(3.1-6)ln[x-Aronln(l+r)(3.1-7)ᡈlog[^ߟ]x-Arn(]=--------------(3.1-log(l+r)7)*x[(l+r)-1]r(l+r)H(3.1-8)ÛÕAÜ°Ḅᔆ◤⌕Õàá☢Ḅãᦪäåæ4(1+r)ᔣ8(4+x)(l+/)+x=0{2.᪷më⍝ìᐜxᔣî ï722ðᐗïòó▲d20039ᨴ-20139ᨴᐳ120óöᵨ÷øjùòᨴú2338ᐗ.ûüýù16óùᒕ104óïòÿ 198155ᐗᐜᜮᨵ5ᐗᵨᵭ5ᐗ.5ᐗ.ᑮᢇ ᐜ !ᢝ#$▲&'ᓽ)*+105$,-ᢥ᯿0Ḅᑭ᳛6.12%4Ḅᨴ6789:;<=⍝?@ᨵAᨴᑭ᳛r78ᡃC✌ᐜ⌕FG4=220000,=120,X=2338.:IJ(3.1-9),ᓽ:220000(1+r)120+1-(220000+2338)(1+r)120+2338=0ᡃCKᑭᵨMathematicaᦪMNOᩭF:.✌ᐜQR(3.1-9)TUḄVᦪWClear[aO,f,n,r,x]
48f[aO_,n_,x_,r-]:=aO(1+r)A(n+l)-(aO+x)(l+r)An+xXKᓫZ“File”[ᓫ\ᐝ᪗_ᑮ“Palettes”⌱⚗ᙠcḄd[ᓫ?)ᓫZ“BasicCalculation”⚗ᢥefghḄijkl⌱mnoKpqrᐭKWᦪMtuḄvu£aO,n,*,r:aO1rp1aOx1rnxf[aO,n,x,r]4(l+r)+i—(Qo+x)(l+r)"+xᯠxyz{ḄaO,n,x|}~.᪷ᡃCᑭḄ:rḄ'ᓄQᜧ0,0.2.a0=220000;n=120;x=2338;Plot[f[aO,n,x,r],{r,0,0.02},AxesLabel{r,f}]Ḅᜧᙠ0.005▬ᡃCK)rḄ'ᓄ{0.004,0.005},WPlot[f[aO,n,x,r],{r,0.004,0.005},AxesLabel->{r,f)]ᡃCKᵨ0.0042}FfḄ
49FindRoot[f[aO,n,x,r]=0,{r,0.0042}]{r->0.00420197 ᑭᵨFindRoot}QḄ¡6¢£⌕ḄᡠKg¥¦§Ḅ¨©6ª«ᨵᦔḄ.ὃ⚪;°ᔲᵨSolve[f[a0,n,r,x]=0,r]ᡈNSolve[f[aO,n,r,x]==0,r]ᩭFr.³´µ¶·¸¹ᡈὅA»Cᔜ½Ḅ¾6¿-9r«0.00420197,ᡈὅr«0.004202,Àᑭ᳛0.050424.)ᵫÂ3.1-4 ,ᑖÄl&=16Çk=15ÈÉÊᑖÄÈÉ2338A=220000(1.004202)16[(1.004202)16-1]160.004202Ç233846=220000(1.004202)15[(1.004202)15-1]0.004202ᑮḄËᑖÄ;196656Ç198161.=⍝?Ḅ198155@ᨵ┯Í,-198161ÎÏq198155.ÐÑAÒ=⍝ᨵÍ.Ó▭gᐜÕ15$ᨵ105$⌕.hᙠḄ40=148,155,H=105,ᑭᵨ3.1-6 ᢥ᯿0Ḅᨴᑭ᳛r=0.0051ÈÉ4Ḅᨴ61825.86.4&5ᐗ*+105$ḄÖ4Ḅᨴ62442.06.MathematicaᨵᐶØḄÙὅK¦W☢Ḅὃ⚪Ûgᡠ¥ᡃC°ÜᵨÝÞব1-3 ᑮÂ3.1-9 ḄÖᡃCK:àá7âᐵḄä⚪.å⚪A 1.&6æ çèᨴᐜᑭ)éNæᑖḄjêN ᦪMÝÞëìí᪵92.ïðḄñᵨᓱóô 12,000õᐗöðḄᑭ᳛19.9%/À.ᑭ6ᢥᨴÈÉḄ.Q¿-᪵Ḅᨴpõᐗ÷°ᙠa.2ÀᎷQ&ìᨵ0Ḅñᵨᓱùú.b.4ÀᎷQ&ìᨵ0Ḅñᵨᓱùú.ó.hᙠᎷQïèᨴᵨñᵨᓱùú105õᐗ.Q¿-᪵ḄᨴPõᐗ÷°ᙠa.2À
50b.4Àó.ὃû⚪üý#þÿ ᙠ10ᨴḄx=3000ᨵ⚪ ᑭ᳛r=5%,ᦪ%=10ߟ15. ᦪ Ḅ!"#$&N=10,'()*+,-./0123/45$1.005/°=1.81940&N=15,'()*+,-./0123/45$1.005 18°=2.454090#ᫀᑖ89:270220ᡈ264000—270000,100 355511/ᡈ36000,150 :<=3ᵨ?)@()A,-⌕CḄ D:1.8194«1.8,0.4<0.8/1.8=0.44444...<0.5,0.44x600000=264000,0.45x600000=270000.2.45409«2.5,1.5/2.5=0.6,0.6x600000=360000.ẆFG⚪1.ᵬI*J.KL60000Mᐗ ᑭ᳛:1.2%,25O.ᎷQAᨴRS/ᓽ*ᨴ:*U0 Vᨴ⌕12Mᐗ3/#ᫀ$ 632Mᐗ.WᑭX:189600Mᐗ.0YZᨵ[*J.KL\ᩭ^ ᩩ`ᨵᓄ bcde'4f2O g⌕$1.VhJᨴi*j=kᩭḄ*h lᨵmn'Ḅopq2.D:VhJᨴb⌕r's*tᦈv ᦻxyz1{ ⌕C'⚜}3JᨴḄ./ᓽ632x3=1896Mᐗ ~2ḄWS:15168Mᐗ 1896gA15168Ḅᑖ*.ᨵ:Aᔠ)Ḅ0Y[*J.KL/᠄ᗐ᪀!0Aᯠcd3kᩭḄ*U/*Jᨴ0ᑖ:RḄU I~xx/1,rr/24fOᔩ3&A 4f12ZO3kᩭḄ*U/*Jᨴ0ᑖ:RḄ0U I~xx/m,rr/4fOᔩ3m8Z᪵3জn149155,¡ᜧL'HospitalᡈUHopital !ᑣ oo•00¤▲.2.:<=§᪵Ḅ.ᑭ᳛ W/WᑭX0ᨵ¨§ᕖ3ª«▅+☢Ḅ199812ᨴ30¯°±○³´µḄ´⍝$“*¸WS:13.5¹ᐗḄJº»¼½ᔠ ᙠ¾¿À)\{ÁÂ& ~ÃSÄŹᐗdÆ YÇ✌jᔣ¿À.ḄÊËÌÍ\ÎKLÏÐÑᝣÓÔ{ᾯÖ.v×Ø ÙÏᡭ)8¹ᐗKÛ±Ü5.5¹ᐗᖪÞឋ ᑖ8fàḕ ÀâãäÀÜå À¼ᙢçèéêë ᐸÂ&A Y13.5¹ᐗ,
51ᑖ15O ᙠᑭ᳛§Ḅîï+ḕ ÀVᨴ⌕Cð}X1175.46ᐗ/ᐸñKÛ±660.88ᐗ ᖪÞឋ514.58ᐗ0 ~å ÀVᨴ⌕C1116.415ᐗ/ᐸñKÛ±634.56ᐗ ᖪÞឋ481.855ᐗ0.ᢥ180Jᨴ*) ḕ ÀḄóᙠå À⌕110628.1ᐗ.ô¾¿Àᙳö Â&¨*᪵÷ãøù.ᨵᐵÀ¾ᔣûὅýþ^ ḕ ÀÿᯠḄᐜᑣᵨᢥᨴᨴᨴ ᪵. !ᓫḄ#$%ᨴ1,000ᐗ' ᨴᑖ)*100ᐗஹ900ᐗ,'-ᨴᑖ).*200ᐗஹ800ᐗ/012.,34Ḅ56738ᳮ:Ḅஹஹ;.*<38✌>?☢ABᑭ*DEFG ᑭHᙳJHKᑮᨴM.NOPQR;ST8UVWḄ.JX YḄZ[\ᝣḄ^_`abGcḄde*fThijkl;cdmnoGThijḄᑭkRp;.q1999r1ᨴ1sb◀uᶇᨴw;y2zGᑭ{|Ḅ⌴~;ᨴ=j+j>ᨴᦪ+`kXᨴᑭ᳛. jᢥPQp;kRw☢Ḅ⚪1.ḕ4Ḅ“ᨴw;ᐜᑣrMḄᨴ᪵RzᩭḄ2.c2zḄ“ᑭ{|w;”Ḅᨴ᪵RzᩭḄᵨ34ḄkR.3.34Ḅ“ஹஹ;”᪵XᑮḄ4.¡ᑖ᪆£QR;Ḅ¤ᜐ¦ᑭ§.ᙠ©ὁu◅ஹᳮ¬⚪MᨵV®1¯Ḅ⚪.ᙠজpp291295“'£⁚ ▤³ឋµᑖp[”¶ᐭ¸#$¹º»¼j ½¾¿Àµᑖp[¿ÀÁÂᦣ¿ÀḄᐵÅᡃǸij*#ᩭn⚪ᎷÉ ÊËḄᢗÍᡈÏÐ*A ,ᓫÑ?ÒḄᑭ᳛Ð*ÓÔ?ᎷÉ?Ò½¾Ḅ,A⌕×AᓫÑ?ÒᡠÙÐ*%ᦋ*t>0?ÜᡠÙA[%,%+â/]NᡃÇᩭ4Ý¿Àᐜὃ⇋.ᙠ?ÒàÒ%+Aå?ÜᡠÙ*AQ+(% ,%?ÜᡠÙ*AQ ,
52æ0ᙠàÒN+çᡠÙḄèé*4(%)A%,êEᨵA(t+Ar)-A(r)=M(r)ArᡈAa+Ar)-A(O----------------------------M(0ArN/+âḄîïâ/ðᩭðZñ7ò?ᓽâ-?Xᑮᑡ½¾¿À(µᑖp[¿À)dA(t),ø--------=rA{t}t>053dex=0,x-"------=enᑣᨵ#dr2reer=1+r+2!+r,-.ᑣ/012ᨵ345 47)84(S;ᙠᩭὃ⇋@ABCᓽᓫFGHIBJKḄA%,LMNOᡂA(?+A?)—A(t)—rA(?)A?—xA?:U-0,VWᑮdA{t}“ஹ[----=M(0-xt>054-rtd40-rre--------reA{t)=--------------=-xertdtdtcoᑮtdᑖVWᑮx*g4(—40)=———1)rYA«)=M+—i—l)e"r=M+±(i-/)rxx=(4——)e"+—rrrkot=kGpᑭᵨeḄ345 erk«(1+r)kVWᑮA(Z:)=(4--)(l+r/+-rr=A)(l+r/--((l+r/-l)r#=A()8ᑣtuMNvwxḄᑖ2XXX0=4enr+-(l-enr)=(4——)enr+-
55X—(enr-l)log[^ߟ]x(enr)n=—"34ᔆ4=enr-l24=°Ḅr,◤⌕☢ḄᦪAArern-xern+x=00;ᙠᡃ01v3ḄᦪᩭᓽA)=120,r=0.0042,X=2338.ᡃ0r2220000,=Clear[aO,n,r,x]g[a0_,n_,r_,x_]:=a0rExp[nr]-xExp[nr]+xg[a0,n,r,x]rnrn.a^re-xe+xa0=220000;n=120;x=2338;Plot[g[aO,n,r,x],{r,0,0.01)]FindRoot[g[aO,n,r,x]==0,{r,0.0042}]{r-0.00423133).r§¨20.00003133.
56ᵫL-1)22000/2/-320r\e-1r=0.00423133;930.8926•Exp[0.5077596]Exp[0.5077596]-l2338.ᐸªᶇ¬⚪.ᡠ0ᙠ°±ஹ²ᳮ´µ¶Ḅ·v3K᪡¹º»tuMN¼½ᑖMN2.᧕¿ÀÁ⚪38ÂÃÄᦋOÆ/Ç/È᧕¿ÀḄÉÊᙠজpp.166-176"Ë⁚ÌᦪḄÎᨬᜧஹᨬ.ΔvÑ“8ஹᨬᜧÎᨬ.ÎÁ⚪”ᦋ2“8ஹᨬÒᓄÁ⚪”ÔÕᐭ“×᧕¿ÀÁ⚪”.ØᓄᎷÚÛ᧕¿ÀᵨᩞḄÝdᐸ⊤☢dᡂß,.ØᓄMN1ᑖ᪆áᎷÚÛ✌ᐜÑäᧇÀ5 æᡂ3ÂçᙊéÝᨵ3KᔠᳮឋḄ.⌕äᧇÀᑁÝd3KGíî᧕¿Àᑴ¬ᡠᵨḄᩞᧇᨬḕḄ⚔òḄçóác⚔òᑮôõḄö,.÷▭ùᵨ"úûüᩭ⊤ýVÛÝdþKḄçᙊéÝᐸ⊤☢dᨬ.Ḅÿ ⊤☢ᵨS⊤ᵨ⊤ᑣᨵSr,h =27rrh+71rl+Tir1-27r[r2+rh\Vdh,h=V/7Tr2.ᡃ Ḅᦪ !"
57min5(r,/z)r>0,h>0s.t.g(r,h)=0ᐸ%s&᪗(ᦪg(r,/z)=V-7Vr2h-0)*ᩩ,./0Ḅ(ᓽ3ᑁ56)ᓽ⌕ᙠ56Ḅᩩ, :3Ḅᨬ<Ḅr,=.?ὃ⇋ᩞᧇDEḄF?ᎷHᡠᵨᩞᧇJ3Ḅ⊤☢ᡂLMNOᐸ%PQ☢ḄRS "F={AbsoluteThickness[1],Line[{{-3.2,12.4},{-3.2,0},{3.2,0},{3,2,12.4},{-3.2,12.4},{-3,12.2},{-3,0.2},{3,0.2},{3,12.2},{-3,12.2}}]}mygrapg=Show[Graphics[F],AxesLabel->{x,y},AspectRatio->Automatic,PlotRange->{-1,12.9}]F={AbsoluteThickness[1],Line[{{-3,0.2},{-3,0},{3,0},{3,0.2},{3.2,0.2},{3・2,12.2),{3,12.2),{3,12.4]，{-3,12.4},{-3,12.2},{-3,2,12.2},{-3.2,0.2},{-3,0.2},{-3,0},{3,0},{3,0.2},{3,12.2),{-3,12.2},{-3,0.2},{3,0.2}}]}mygrapg=Show[Graphics[F],AxesLabel->{x,y},AspectRatio->Automatic,PlotRange->{-1,12.9}]
582+h=YIᐔ-TᐭS(r,h),VᑮVV9+r——-]=29S(r)=X[ZH-----]7ir~7ir:[\(]^\,criticalpoint)0=S'(r)=2%(2—`)=g(2r3--)ᐔrr7ibᵫ2VS(/0=2»(2+1)\ுO/ுoᡠᵫজ141—149"gh⁚jk(Taylor)lm”ᱯpp.142Ḅ
59/W=/(᳝))+:(%)(x-/)+-᳝))2ম0⌼“`0⍝5~<\▭ᐰ<\,]^\5Ḅᨬ<☢5%=6,V=6%2V2ᐔᨵᨵḄ᧕3ᔩᨵᓄ!2ᑖ᪆ᎷH"ᵨ᥎5 ⚔¡ᑮḄ¢E⌕Mᐸ£Ḅᩞᧇ⌕¢D⌕¤¥ᎷH◀᧕3Ḅ⚔ஹ¨3ḄDEª«¬®⚔ஹ¨ḄDEª«¯ᓃ.±²5 ¢E³ᙠ«᪵ᩞᧇḄDE(µ☢Ḅ).¶ᡃ·¸ Ḅᦪ .¹º»¼ὃ⇋ᡠᵨᩞᧇḄ.F={AbsoluteThickness[1],Line[{{-3.2,0},{3.2,0},{3,2,12.8},(-3.2,12.8}{53.2,0},{3.2,0},{3,0.2},{3,12.2}{-3,12.2},{-3,0.2},{3,0.2}}]}mygrapg=Show[Graphics[F],AxesLabel->{x,y},AspectRatio->Automatic,PlotRange->{-1,12.9}]áâãäåᦪ"Hæᧇ3Ḅr¶d=2r,3Ḅh.3ᑁV.b◀⚔ஹ¨ᓽè☢ḄᩞᧇḄDE.ᐸ%é=
60êãäᡠᵨᩞᧇḄsrãäëí6åᦪ0î6åᦪ.æᧇ3è☢ᡠᵨᩞᧇḄS(r,h)-(7T(r+40ᐔ )h1æᧇ3⚔ᡠᵨᩞᧇḄabᐔr1æᧇ3¨ᡠᵨᩞᧇḄabrcrᡠᑖpS(V,/z)=7ib(2r+b)h+27ia(r+b)2b=2ᐔrhb+l/car^b+^Tirab1+hjib1+Ijiab3V(r,h)=7rrh5ᡠïḄ⚗ñᶍ(ᐸó⌕ḄᔠᳮᎷHᡈᓄ÷O).¶5V(r,/z)xS(r,h)-2/irhb+27rar2b¬g(r,/z)="r2h-Vᡃ Ḅᦪ !"minS(r,/z)r>0,h>Qs.t.g(r,/z)=0ᐸ%s&᪗(ᦪg(r,h)=O)*ᩩ,P/0Ḅ(ᓽ3ᑁ56)ᓽ⌕ᙠ56Ḅᩩ, :3Ḅᨬ<Ḅr,=a¤V
61r,høäù?úᔠ.¹5~:ᩩ,Ḅû⚪.!Ḅ:ý"5þýÿ ᓄ ᩩ⚪ᐗᦪḄ ᩩ⚪g(r,h)=^rh-V=0h=V/7T/ ᐭs, ⚪ᓄ!d.•$ Sᨬ&ᓽr 2VSr,/zr =92b[-----F2/iar]rᨬ&.()*!+ᐸ-ᦪ./3=20,2"—=]67 28"3—V-=o.2ar=ad.9ᦪ:ᜧ<$Zr=2,ᓽ=>?a=2,ᓽ⚔ஹBCḄDEFᐸGᩞᧇDEḄ2J.KLMrNO SPᑮ&RSSḄ6▤-ᦪV5=4ᓡ»+X]ு0,• •r>0.rᡠ[MrNO SPᑮ\]&^()*_ᨵ^abFᐰ\&.d⚪ ᡈfὃ⚪ijkᶍmno&piqrs᪵uSV,h =^-r+/? 2-7rr2 h+2a7rr-\-b 2b=b[7rh2r++2aᐔ1-+/? 2]
622VV(r,/z)=nrh,h=——rnr ᐭhḄ⊤Pw xS/ᑮS")…+y+2z{ ᔆ+}2,2r2/7(r+Z?)r___-i_=-----------V+2a?ir33=0r——/~᪵Ḅpi!ᙠ aᑖᐝᦪ/(ᐗ)Ḅᨬᜧ⚪max/(x)xe[a,h]ᨵḄᦟᩞFrederickR.Adler(DepartmentofMathematicsandDepartmentofBiology,UniversityofUtah)ModelingtheDynamicsofLifeCalculusandProbabilityforlifescientists,Brooks/ColePublishingCompany,1998.(20052᱐)ᦪᨬᜧஹᨬ&ḄS!p.200,Algorithm3.1(Findingglobalmaximaandminima)1.Computethevalueofthefunctionattheendpointsandanywherethefunctionisnotdifferentiable.RSᦪᙠ*j*
63ᜐḄ.2.Findallcriticalpoints.ᐰ]()*.3.Computethevalueofthefunctionatallcriticalpoints.RSᡠᨵ()*ᜐḄ.4.Thelargestofthenumbersfoundinstep1and3istheglobalmaximum.Thesmallestofthenumbersfoundinstep1and3istheglobalminimum.1ஹ3/Ḅᨬᜧ &FᦪḄ ᐰ\ᨬᜧ &.p.202,Algorithm3.2（Findinglocalmaximaandminimawiththesecondderivative）1.Findallcriticalpoints.2.Computethesignofthesecongderivativeatallcriticalpoints.3.Criticalpointswherethesecondderivativeispositivecorrespondtolocalminima,andcriticalpointswherethesecondderivativeisnegativecorrespondtolocalmaxima.M¡¢£ᡃ¥¦⚪Ḅfὃ:¨©FSuS¨©ª⌕u¬SᙠO▭®F¯Ḅᔩu/ᔣ~²³´.S（algorithm）µ¶RS·¸Ḅ¹º»ᢣ+ ½M·¸b¾S（algorithmic）·¸¿ÀÁ? µḄSḄµᦪḄ¡ᭊᐭḄÄÅÆᐭ（input）,½ÇᐸÈḄᙠ?/ᑮÉᐰᵫÆᐭᢣ+˵Ḅpi（result）（ᡈÆ（output））.-Íᦪ²Î0ᐰ}ᔁ1,0²᱐Ð1994,pp.119121.ÑᓄÒÓ3᧕ÕÖ[ÑᓄᙊØÙᙊÚÛÖ ÜÝÞEßàáâᡂäåᡃ¥ᎷçᙊØ]ᑖFèᙊØO▭¿b¡Féêë ìêëḄíîëïð½/ḄᙊØ.ᎷçᡠᵨᩞᧇòÖᑁḄ⊤☢õᡂö÷ᓽᔜ]ᑖḄᩞᧇõò]ᑖḄ☢õᡂö÷ᩭúûᑴÖᩞᧇḄõýþ þ÷ÿᜲ!
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702᪀ᡂᵫᎷ4iᎷ5,f»ᑮ]Gcd\ᙠ¼¾¿Ḅ=ÁµkjlᵨḄ2{dX(t)/dt=aX-bXYEdY(t)/dt=cXY-eY)ᐸwc=ab.(1)Â!•WÃÄឋ³ÆZÇ]N>WᦪXÈḄÉᓄÊËᡃD7l°ឋᑖ᪆ਮ{aX-bXY=OHcXY-eY=O}h»ZÇ(1)Ḅ>W⊝¥Í0(0,0)'M(e/c,a/b)=ª(1)Ḅ>Î[wÏÐdt,ᑖÑÉÒN»✌Ó ᑖF(X,Y)=cX-dln|X|-aln|Y|=k(2)p᧕Nr'ᦪF(X,Y)ᨵÔ"ÕÖM(e/c,a/b)×ᵨḄᐙᑖᩩᑨÜᩩᑨÜM!FḄÝÖÞ᧕ßX-8(B7A឴áᐭâ)ᡈÈ-8(A!ᙽåy:Ḅᩈᜮ)ᙳᨵF-8EX-0(Al]ÉéêëB7ìí¼■ᜓ)ᡈY-0(A¼ëð᳛:)ñᨵF-8,ᵫBòrᙠ•ó▲ᑁõö÷Ḅ'ᦪz=F(X,Y)Ḅøé!MùÝÖúᙠᓶ▲ᔣý¼▲þÿḄ☢z=k(k>0)Ḅᙠ☢XOYḄᢗF(X,Y)=k(k>0)MḄἕᡂ Ḅᢣᦪᡂᕜ$ឋ&ᓄ)*+,-..12345᧕ᳮ+Ḅ8A9Ḅ:ᡂ Y(t);<=B? A9 X(t)3ᓣABA9CDEFGHIJ,:ᡂ L)N3ᓣOBBCN A9EFOᩭQ X(t)N;71ὃ⇋ᑮ3Ḅ=Ḅ ᡂᦪlh,hOḄᵨ¡ᙠ¢£¤;3 Ḅ¥&l{dX/dT=aX-bXY-hX=(a-h)X-bXYAdY/dt=cXY-eY-hY=cXY-(e+h)Y)(4)¦(4)(1)§Z¨nὅª«ᐰ`ὅ®¯(1)°XLḄᦪᑖgᣚᡂOa-he+h²(4)ᨵx'=(JXdt)/T=(e+h)/c,y'=(JYdt)/t=(a-h)/b(5)ᑭᵨ(5)ᡃYZ¨ᵨ¡hḄµᜧ·X'µ¸L'º»ᡃYḄ¼½ὃ¾$VᵫB¿ÀÁ·yº»ᓽhºÄ=¦ᨵᑭBᡂ LḄµÅCVolterraÆᳮ²ÆᳮᵱÈᨵḼÊ⌕ḄᢣÌ12JᜧḄ᰿£ᨵ=_hÎÏÐᑮÑ1Ḅᦔ*_ᑭḄᡂÅᨵ=Ó᠒᠒ÕÖ᠒᠒O+׸3〉8ḄÙÈḄ C?᠒᠒<Ú:ᡂ Û_?<ÚÜ°ÞßᔣÙáâᦪ¼ᢣãsäåḄ•ᱯÎèlOéᱯÎêḄëì,íÊ⌕Ḅîᓄ Ꮇᵨ〉8Ḅᦪðᐹyᑮhoᦪ)᪀ᵨᩭ+,ᱯÎãóḄãsឋ⚜õöḄ÷ᩭø¤ùúᜐᳮèḄᓄüý þᑴ ◤⌕Ḅᩭᦪᵨ᪾ ᩭ⊤ ᦪ!ᙠ#▭%ᵨḄ◤&'(Ḅ ⌕)*#▭+⚪-.◤/ᦪ0 123456ᦪ7ᦪ᪵ᨵ:ὁᔊ=> ?@AB@C-!D:ὁḄᦪ0 EFGᨵHIJKL!ᦪḄDᐝNᐺQᜩ ᦪSTUḄVW7XWᔣᐸ[\ᢈ^⚞`ab cde%ᵨᦪḄ⚞`fᙠghiᔣJjᓄ ᦪjᓄ ◤/ᜧjḄᦪ0ᱯn!oᢈ^ஹoqrstᐶv wxḄyz7V{%ᵨ ᦪᙠ|}~oᢈ^5vḼᓝᑖᐵḄᵨ2ᦪ⌕Ḅ34
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