习题八习题解答与问题

h=0.2)6.00(1)0(2≤≤⎩⎨⎧=−−=′xyxyyyf(xy)=–y–xy2yn+1=yn+h[–yn–xnyn2]yn+1=(1–h)yn–hxnyn2y0=1y1=0.8y2=0.6144y3=0.4613)21(2)1(38≤≤⎩⎨⎧=−=′xyyyh=0.2f(xy)=8–3yyn+1=yn+0.5h[(8–3yn)+(8–3yn+1)]h=0.213161371+=+nnyyy0=2y1=2.30769y2=2.47337y3=22.56258y4=.61062y5=2.63649)5.11(5.0)1(112⎪⎩⎪⎨⎧=−=′xyyxyxyh=0.1xxxy+=1)(nnnnnxyyhyy)1(~1−+=+])~1(~)1([5.011++−+−+=nnnnnnnnxyyxyyhyyy0=0.511.11.21.31.41.5yn0.50000.52380.54550.56530.58340.6001y(xn)0.50000.52380.54550.56520.58330.6000yn–y(xn)00.2570×10-40.4503×10-40.5962×10-40.7066×10-40.7902×10-4⎩⎨⎧==+′1)0(0yyy451.0=h)2.0(y510−][211+++−=nnnnyyhyynnyhhy+−=+221y1=0.9048y2=0.8186⎩⎨⎧==′axyyxfy)(),(0)3(211−+−+=nnnnffhyy])()[(1),(11nnnnfxxfxxhyxf−−−+−≈nxxnnxxnxxfdxxxfdxxxhdxyxfnnnnnn∫∫∫+++−−−+−≈111)()([1),(11]2321[1nnffh+−=−)3(211−+−+=nnnnffhyy⎩⎨⎧=−=′1)0(yyy)()22()(nhxhhynhynn=+−=≈0→hxe−][211+++−=nnnnyyhyynnyhhy+−=+221nnnhhyhhy)22()22(0+−=+−=46hnhhhnhhhh+++−=+−2222)221()22(hxhhnhh+++−=2222)221(110)1(lim−→=−exxxnnxhxhhhnhehhhh−++→→=+−=+−222200)221(lim)22(lim0→hxe−⎩⎨⎧=−=′2)0(38yyy2.0=h)4.0(y⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧++=++==+++=+).43,43(),2,2(),,(),432(9231213211hKyhxfKKhyhxfKyxfKKKKhyynnnnnnnnO(h4))(),(),()(nnnynnxnxyyxfyxfxy′+=′′2)]()[,()(),(2),()(nnnyynnnxynnxxnxyyxfxyyxfyxfxy′+′+=′′′)(),(nnnyxyyxf′′+)(),(1nnnxyyxfK′==)(),()(8)(8)(2)(3222hOyxfxyhxyhxyhxyKnnynnnn+′′−′′′+′′+′=),()(243)(43)(23nnynnnyxfxyhxyhxyK′′+′′+′=)(),()()43(21)()43(21322hOyxfxyhxyhnnynn+′′−′′′+)()(61)(21)()432(9432321hOxyhxyhxyhKKKhnnn+′′′+′′+′=++47)()(61)(21)(4321hOxyhxyhxyhyynnnnn+′′′+′′+′+=+)()(61)(21)()()(4321hOxyhxyhxyhxyxynnnnn+′′′+′′+′+=+11132)4(31+−+′+−=nnnnyhyyy1111132)4(31)()(+−+++′−−−=−nnnnnnyhyyxyyxy)()(21)()()(321hOxyhxyhxyxynnnn+′′+′+=+)()(21)()()(321hOxyhxyhxyxynnnn+′′+′−=−)()()()(21hOxyhxyxynnn+′′+′=′+)(32)(32)(32)(31211hOyhxyhxyhyxynnnnn+′−′′+′=−+++)(32)(32)(31111hOyhxyhyxynnnn+′−′=−++++baxy+=′y(0)=0bxaxy+=221yn+1=yn+0.5h[f(xnyn)+f(xn+1yn+hf(xnyn))]yn+1=yn+0.5h[(axn+b)+(axn+1+b)]xn=nhyn+1=yn+0.5h[a(2n+1)h+2b]=yn+0.5a(2n+1)h2+bh)()12(5.010210101NhbnahyyNnNnnNnn+++=∑∑∑−=−=−=+y(0)=0yN=0.5ah2[(N–1)N+N]+b(Nh)yN=0.5ah2N2+b(Nh)=0.5a(xN)2+b(xN)248nnnbxaxy+=221nnnbxaxxy+=221)(])13()3[(41)1(1111−+−++++=−−+nnnnnfbfbhbyybyb=11≠by(xn+1)–yn+1=y(xn+1)–[(1–b)yn+byn-1+0.25h(b+3)fn+1+(3b+1)fn-1]Taylor)()(!31)(21)()()(4321hOxyhxyhxyhxyxynnnnn+′′′+′′+′+=+)()(!31)(21)()()(4321hOxyhxyhxyhxyxynnnnn+′′′−′′+′−=−)()(21)()()(321hOxyhxyhxyxynnnn+′′′+′′+′=′+)()(21)()()(321hOxyhxyhxyxynnnn+′′′+′′−′=′−y(xn)=yny(xn-1)=yn-1)()()1(31)(4311hOxyhbyxynnn+′′′+−=−++b=11≠b)10(1)0(20≤≤⎩⎨⎧=−=′xyyyh=0.1h=0.2f(xy)=-20yK1=-20ynK2=-20(yn+0.5×hK1)=-20yn+200hynK3=-20(yn+0.5×hK2)=-20yn+200hyn–2000h2ynK4=-20(yn+hK3)=-20yn+400hyn–4000h2yn+40000h3yn-yn+1=yn+h(K1+2K2+2K3+K4)/6=(1–20h+200h2–4000h3/3+20000h4/3)yn1.0=h2.0(1–20h+200h2–4000h3/3+20000h4/3)=0.3333(1–20h+200h2–4000h3/3+20000h4/3)=5h=0.1h=0.21∫badxxf)()(tfy=′y(a)=049∑−=≈10)()(nkkhtfbyh=(b–a)/ntk=a+kh(k=01n)yk+1=yk+hf(tk)(k=01n)∑∑∑−=−=−=++=1010101)(nkknkknkktfhyy∑−==≈10)()(nkknhtfyby21926(vanderPol)′′+−′+=yyyyµ()210µyyyyyy1233==′+−,(µ)yy′=′1yyyy′−+′′=′)1(22µ0)1(2=+′−+′′yyyyµ⎩⎨⎧−=′−−=′1213121)3/(yyyyyyµ3(Dawson)fxxtdtx()exp()exp()=−∫220′fx()fx())(xfy=∫−=xdttxxf022)exp()exp()(1)(2)exp()exp()exp()exp(2)(22022+−=−+−−=′∫xxfxxdttxxxfxf(0)=0y=f(x)⎩⎨⎧=−=′0)0(21yxyy450π≤≤−=+′′−xxxexuxux0),cos2(sin)()(0)(,0)0(==πuun1+=nhπxj=jh(j=01n+1)f(x)=ex(sinx–2cosx)21111)()(2)()]()(10)([121hxuxuxuxuxuxujjjjjj−++−+−=′′+′′+′′–(12–h2)uj-1+(24+10h2)uj–(12–h2)uj+1=h2(fj-1+10fj+fj+1)(j=12n))()()(111−−−=+′′−jjjxfxuxu)()()(jjjxfxuxu=+′′−)()()(111+++=+′′−jjjxfxuxu1/1210/121/12)10(121)10(121)2(11111112+−+−+−++=++++−−jjjjjjjjjfffuuuuuuh–(12–h2)uj-1+(24+10h2)uj–(12–h2)uj+1=h2(fj-1+10fj+fj+1)5Adamas),(yxfy=′y(x0)=y01yn+2=yn+1+h[3f(xn+1yn+1)–f(xnyn)]/22yn+2=yn+1+h[5f(xn+2yn+2)+8f(xn+1yn+1)–f(xnyn)]/121[xnxn+1]f(xy)])()[(1),(11++−+−≈nnnnfxxfxxhyxf∫∫∫++++++−+−≈++212121)(1)(1),(11nnnnnnxxnnxxnnxxdxxxfhdxxxfhdxyxfx=xn+1+th2/)(2102121htdthdxxxnnxxn−=−=−∫∫+++2/3)1()(210221hdtthdxxxnnxxn=+=−∫∫++5112321),(21++−≈∫++nnxxhfhfdxyxfnn),(yxfy=′∫∫++++=′2121),()(nnnnxxxxdxyxfdxxy2/)3()()(112nnnnffhxyxy−≈−+++yn+2=yn+1+h[3f(xn+1yn+1)–f(xnyn)]/22[xnxn+2]f(xy)])()(2)([21),(22112+++++−≈nnnnnnfxBfxBfxBhyxfBn(x)=(x–xn+1)(x–xn+2)Bn+1(x)=(x–xn)(x–xn+2)Bn+2(x)=(x–xn)(x–xn+1)x=xn+1+th6/)1())(()(3103212121hdttthdxxxxxdxxBnnnnxxnnxxn−=−=−−=∫∫∫++++++3/2)1())(()(31023212121hdtthdxxxxxdxxBnnnnxxnnxxn−=−=−−=∫∫∫++++++6/5)1())(()(3103122121htdtthdxxxxxdxxBnnnnxxnnxxn=+=−−=∫∫∫++++++]85[61])(2)([123221121nnnxxnnnnnnfffhdxfBfxBfxBnn−+=+−++++++∫++),(yxfy=′∫∫++++=′2121),()(nnnnxxxxdxyxfdxxy12/)85()()(1212nnnnnfffhxyxy−+≈−++++yn+2=yn+1+h[5f(xn+2yn+2)+8f(xn+1yn+1)–f(xnyn)]/121x⎩⎨⎧=+=′0)0(0yxbaxybxaxxy+=25.0)(n=nhyny(xn)–yn=0.5ahxn231.fxedttx()=−∫20[,]03352(1)dydxfxyxy==⎧⎨⎪⎩⎪()()00(2)dydxfyyxy==⎧⎨⎪⎩⎪()()004y(n)+any(n-1)+an-1y(n-2)++a0y=f(x)y(x0)=α0y(k)=αk(k=12n–1)5⎩⎨⎧−=′=≤≤=+′−′′5.0)0(,0)0(30sin22yyxxeyyyx53