东北师大《常微分方程》习题解答

121.21(1)xdxydy=1222121cxy+=cyx=-22(2)yydxdyln=1,0==yy1,0¹¹yydxyydy=ln0ln,lnln11¹=±=+=cceeeycxyxxcxceey=(3)yxedxdy-=dxedyexy=ceexy=-(4)0cottan=-xdyydxxydxdycottan=0=y0¹ydxxxdyyycossinsincos=.11cossinln,coslnsinlncxycxy=+-=,0,cossin1¹=±=ccexyc2(1)1)0(),1(=-=yyydxdy1,0==yy1,0¹¹yydxdyyy=--)111(0,1,1ln11¹=±=-+=-cceeeyycxyyxxc1)0(=y0=c1=y(2)1)0(,02)1(22==+¢-yxyyx0,1222=--=yxxydxdy0¹ydxxxydy1222--=3cxy+--=-1ln121)0(=y1-=c11ln12+-=xy(3)0)2(,332==¢yyy0=y0¹ydxydy=323331)(,cxycxy+=+=0)2(=y2-=c3)2(-=xy0=y(4)1)1(,0)()(2222-==+-+ydyyxxdxxyy0,0==yx0,0¹¹yx01122=+-+dyyydxxx0,,ln1ln1111111¹=±==-++---cceeeyxcyyxxyxyxc1)1(-=y2--=ecyxeeyx112---=401122=-+-dyxydxyx)11(1),11(1££-±=££-±=xyyx1,1±¹±¹yx01122=-+-dyyydxxx)0(1122=-+-ccyx6x(x)(1,2).)(xyy=),(yx)('xXyyY-=-x'yyxa-=40'-=--xxyyx2)1(,'=-=yxyy0,¹=cexycce=2c=ln2,2=xy7142122341054104´tq(t),0=kkqdtdqktceq=t=00q,kteqq0=102)4(qq=0402qeqk=42ln=k042ln1201208)12(qeqeqqk===2450430104)5(,10)3(´====kkeqqeqq24ln=k4024ln3030108)3(====qeqeqqk301025.1´=q1.31(2)0)2(22=+-dyxdxxyy2)()(2xyxydxdy-=xyu=22uudxduxu-=+)1,0()111(¹=--uxdxduuuxcuu1ln1ln=-111-=xcxcu0,)(==-ycyxyx5(4)xyxyyxtan=-¢xyxydxdytan=-xyu=uuudxduxcossintan==)0(sincot¹=uxdxuducxu=sincxxy=sin(5)xyxyxyyx++=-¢ln)(xyxxyxydxdy++=-ln)1(xyu=)1ln()1(uudxdux++=1,0-¹¹uuxdxuudu=++)1ln()1(cxu=+)1ln(cxxy=+)1ln((6)yyxyx+-=¢22xyxydxdy+-=2)(1xyu=21udxdux-=)11(12-=-uxdxuducxulnarcsin=xycxxy±==,lnarcsin2(1)0)3()642(=-+++-dyyxdxyx3624-+--=yxxydxdy
=-+=+-03042baba2,1==ba62,1+=+=hzyxzhzhzh+-=24ddzh=uuudduu+-=+124zz)2,1()2)(1(1¹-=--+udduuuuzzcuuu=---z2)12)(2(1,)1()2(23+=--=-xyxycxy(2)0)324()12(=-+-++dyyxdxyx32412-+++=yxyxdxdyyxu+=23255--=uudxdu)1(5132¹=--udxduuu151ln2cxuu+=--xyceyx-=-+212(4)2)12(2-+-=¢yxyy2,1-=+=yvxu2)(2vuvdudv+=uvz=2)1(2zzdudzuz+=+)0()1()1(22¹-=++zududzzzzczzulnarctan2ln+-=12arctan22+--=-xycey30)823()732(2222=-+--+ydyyxxdxyx823732222222-+-+=yxyxxdxydy823732222222-+-+=yxyxdxdy7vyux==22,823732-+-+=vuvududv
=-+=-+08230732baba1,2==ba1,2+=+=hxvuhxhxxh2332++=ddxhw=)1()1(2232±¹=-+wxx)1(1xwwc=-+)3()1(22522-+=--yxcyx1.41.(1)24dyxyxdx+=20dyxydx+=2xyCe-=
.2y=.$y=Ce^{-x^2}+2$.(2)21'2(2)2yyxx-=--1'02yyx-=-(2)yCx=-
.3(2)yx=-.2(2)(4)yxxxC=--+.(3)32ddrrq+=30ddrrq+=3Ceqr-=
.23r=.323Ceqr-=+,332Ceqr-=+.82,.()yyx=,'ky=.(,)xy'()YyyZx-=-.'yxyx-=,'1yyx=-.'yyx=yCx=
.ln||yxx=.,ln||yxxCx=+.3(2)4'20yxyxy++=43'2yyxyx--+=-.$z=y^{-3}$,63dzxzxdx-=.6dzxzdx=.0z=,30y-=,0y=;0z¹,6dzxdxz=,23xzCe=.23()xzCxe=63dzxzxdx-=,23'()3xCxxe-=.2311()2xCxeC-=+,2312xzCe=-,23312xyCe-=-.(4)2(cossin)dyyyxxdx+=-2y--21sincosdyyyxxdx----=-.1zy-=,2dzdyydxdx-=.sincosdzzxxdx-=-.0dzzdx-=xzCe=
.sinzx=-.sinxzCex=-.1sinxyCex-=-.90y=.6.()yx[0,)+¥,lim['()()]0xyxyx®+¥+=,lim()0xyx®+¥=.'()()()yxyxfx+=,lim()0xfx®+¥=,0()()xsxxCfsedsyxe+=0e,1x,1xx,|()|fxe.0101xxxxx|||()|()||+|()|e(x+)xsxxssCfsedsyxeCfsedsedsee+££®®¥elim()0xyx®+¥=.1.51(1)222()0xydxxydy+-=2MNxyx¶¶==¶¶,.233xyyC-=.(2)(2)0yyedxyxedy---+=yMNeyx-¶¶=-=¶¶,.2yxeyC--=.2..(1)22()0xyxdxxydy+++=2Myy¶=¶,Nyx¶=¶,.11()MNNyxx¶¶-=¶¶x,()xxm=.,103222()0xxyxdxxydy+++=.4223364xxyxC++=.(2)432422(22)(3)0yyxyexyydxxyexyxdy+++--=3428261yyMxyexyexyy¶=+++¶,42223yNxyexyx¶=--¶,.14()MNMyxy¶¶-=--¶¶y,41()yym=.,22324213(2)()0yyxxxxedxxedyyyyy+++--=.223yxxxeCyy++=.(3)443()0xydxxydy+-=34Myy¶=¶,3Nyx¶=-¶,.15()MNNyxx¶¶-=-¶¶x,5x-15443()0xxydxxydy---+-=.444ln.xxyC--=(4)3222432(2422)2()0xyxyxyxyydxyxyxdy+++++-+=32344442Mxyxxyxyy¶=++++¶,42Nxyx¶=+¶,.1()2MNxNyx¶¶-=¶¶x,2()xxem=.223222432(2422)2()0xxxyxyxyxyyedxyxyxedy+++++-+=.112224(24)xxyxyyeC++=.1.61..(1)22'0yy-=(')(')0yyyy+-=,'yy=-'yy=.'yy=-xyCe-=;'yy=xyCe=.xyCe±=.(2)38'27yy='py=,3827py=.x2427dppdx=.22712pxC=+.23()yxC=+.,0y=.(3)22('1)1yy+=,1y=±.yt=,21'tyt-=.2211ttdxdydttt==--,2211txdtCtCt=+=----.21xCtyt+=--=
22()1xCy++=.422''(')xyyyxy=-,',''yyy,zdxye=zx.122',''(')zdxzdxyzeyzze==+.,',''yyy2zdxye=,222(')(1)xzzxz+=-,2'21xzxz+=,2()'1xz=,21xzxC=+121Czxx=+11221()lnlnCCzdxdxxCxxx=+=-+.12ln/lnzdxxCxCyee-+==1/2CxyCxe-=,0y=.2.11.:(1)ay=¢(a);(2);2xy=¢(3)yy=¢;(4);12xdxdy-=(5).xdxdy=(1)ayxf=),(,xy,,a.,.(1).(2)2),(xyxf=,y,,kx=±,),(yxf.,,.,,.(2).(3)yyxf=),(,x,ky=(k),,,y0,(3);y0,(4).(2)(1)13(4)21),(xyxf-=,y,,kx12-=,),(yxf.,,.,,.(5).(5)xyxf=),(,y,kx=(k),,x0,(6);x0,(7).2.22yxdxdy-=xoy.,.,..,,.,(8)(3)(4)(5)(7)(6)14,.(8).3.,1.0=h,
=+=1)1(,22yyxdxdy4.1=x.10=x,10=y.,1.11.001=+=xx;2.11.0211=×+=y,2.11.012=+=xx;465.11.065.22.12=×+=y,3.11.023=+=xx;824.11.0586.3465.13=×+=y,4.11.034=+=xx.326.21.0017.5824.14=×+=y2.21.xxdxdytan=(1);0,11:1p££££-yxR(2)44,11:2pp££-££-yxR2.2?(1).1R,yxyxftan),(=2p=y.(2).2R,yxyxftan),(=2cos),(2£=¢yxyxfy.2.?(1)22yxy+=¢;(2)yxysin+=¢;15(3)31-=¢xy;(4)yy=¢.(1)22),(yxyxf+=yyxfy2),(=¢xoy,xoy.xoy.(2)yxyxfsin),(+=yyxfycos),(=¢xoy,xoy.xoy.(3)=),(yxf31-xyxoy0),(=¢yxfy,yxoy.(4)=),(yxfy=
-³0,,0,yyyyxoy,
--=¢0,21,0,21),(yyyyyxfyxxoy,xxoy.3.3123ydxdy=2.2.)0,0(.y3221-=¶¶yyf,x)0(=y,.,0=y(,).dxdyy2331=-16Cxy23232332-=23)(Cxy-±=.0)(³-Cx0=y.)0,0(((9)).0=y,
-£=CxCxCxy,)(,023
--£=.,)(,023CxCxCxy4.2yxdxdy-=0)0(=y:)(),(),(),(3210xxxxjjjj0)0()(0==yxj20121)0(0)(xdssxx=-+=j52022220121])21([0)(xxdsssx