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1、ᐌᖂߚᮍд乬ᐌᖂߚᮍд乬ㄨḜᐌᖂߚᮍд乬ㄨḜㄨḜㄨḜdy=2xydxᑊ∖⒵䎇߱ྟᴵӊ˖[ⱘ⡍㾷㾷˖ᇍॳᓣ䖯㸠ব䞣ߚ⾏ᕫ122xdy=2xdx,ϸ䖍ৠᯊ⿃ߚᕫ˖lny=x+c,ेy=ceᡞx=,0y=1ҷܹᕫy2c=,1ᬙᅗⱘ⡍㾷Ўy=exDŽ2.2ydx+(x+)1dy=,0ᑊ∖⒵䎇߱ྟᴵӊ˖[ⱘ⡍㾷㾷˖ᇍॳᓣ䖯㸠ব䞣ߚ⾏ᕫ˖1111-dx=dy,ᔧy¹0ᯊˈϸ䖍ৠᯊ⿃ߚᕫ˗lnx+1=+c,ेy=2x+1yyc+lnx+1ᔧy=0ᯊᰒ✊гᰃॳᮍⱘ㾷DŽᔧ
2、x=,0y=1ᯊˈҷܹᓣᄤᕫc=,1ᬙ⡍㾷ᰃ1y=DŽ21+ln1+xdy1+y=3dxxy+xy㾷˖ॳᓣৃ࣪Ў˖22dy1+y11+yy1=·ᰒ✊¹,0ᬙߚ⾏ব䞣ᕫdy=dx323dxyx+xy1+yx+x1212222ϸ䖍⿃ߚᕫln1+y=lnx-ln1+x+lnc(c¹0),े1(+y)(1+x)=cx22222ᬙॳᮍⱘ㾷Ў˄1+y˅1(+x)=cx4˖1(+x)ydx+1(-y)xdy=01+x1-y㾷˖⬅y=0x=0ᰃᮍⱘ㾷ˈᔧxy¹0ᯊˈব䞣ߚ⾏dx=dy=0xyϸ䖍⿃ߚlnx+x+lny-
3、y=c,ेlnxy+x-y=c,ᬙॳᮍⱘ㾷Ўlnxy=x-y=c;y=;0x=.05(:y+x)dy+(y-x)dx=0dyy-xydydu㾷˖=,Ҹ=u,y=ux,=u+xdxy+xxdxdxduu+1u+11߭u+x=,ব䞣ߚ⾏ˈᕫ˖-du=dx2dxu+1u+1x12ϸ䖍⿃ߚᕫ˖arctgu+ln(1+u)=-lnx+cDŽ2dy226˖x=y+x-ydxydydu㾷˖Ҹ=u,y=ux,=u+x,߭ॳᮍ࣪Ў˖xdxdx22dux1(-u)11=,ߚ⾏ব䞣ᕫ˖du=sgnx·dxdxx1-2xu-ϸ䖍⿃ߚᕫ˖arcs
4、inu=sgnx·lnx+cy-ҷಲॳᴹব䞣ˈᕫarcsin=sgnx·lnx+cx22ˈy=xгᰃᮍⱘ㾷DŽ7˖tgydx-ctgxdy=0㾷˖ব䞣ߚ⾏ˈᕫ˖ctgydy=tgxdxϸ䖍⿃ߚᕫ˖lnsiny=-lncosx+c.2y+3xdye8:=-dxyy13x㾷˖ব䞣ߚ⾏ˈᕫdy=-+c2ey3e9:x(lnx-lny)dy-ydx=0yy㾷˖ᮍৃবЎ˖-ln·dy-dx=0xxy1lnuҸu=,᳝߭˖dx=-dlnuxx1+lnuyҷಲॳব䞣ᕫ˖cy=1+lnDŽxdyx-y10˖=edxyx㾷˖ব䞣ߚ⾏ed
5、y=edxyxϸ䖍⿃ߚe=e+cdyx-y=edxyx㾷˖ব䞣ߚ⾏ˈedy=edxyxϸ䖍⿃ߚᕫ˖e=e+cdy211.=(x+y)dxdydt㾷˖Ҹx+y=t,߭=+1dxdxdt1ॳᮍৃবЎ˖=+12dxt1ব䞣ߚ⾏ᕫ˖dt=dx,ϸ䖍⿃ߚarctgt=x+c2t+1ҷಲব䞣ᕫ˖arctg(x+y)=x+cdy1=2dx(x+y)ˊ㾷dydtdt1Ҹx+y=tˈ߭=-1ˈॳᮍৃবЎ=+12dxdxdxt2tব䞣ߚ⾏dt=dxˈϸ䖍⿃ߚt-arctgt=x+cˈҷಲব䞣2t+1x+y-arctg(x+y)
6、=x+cdy2x-y-113.=dxx-2y+111㾷˖ᮍ㒘2x-y-1=,0x-2y+1=;0ⱘ㾷Ўx=-,y=3311dY2X-YҸx=X-,y=Y+,᳝߭='33dXX-2Y2YdU2-2U+2UҸ=Uˈ߭ᮍৃ࣪Ў˖X=XdX1-2Uব䞣ߚ⾏dyx-y+514,=dxx-y-2dydt㾷˖Ҹx-y=5=t,߭=1-,dxdxdttॳᮍ࣪Ў˖1-=,ব䞣ߚ⾏(t-)7dt-7dxdxt-712ϸ䖍⿃ߚt-7t=-7x+c212ҷಲব䞣(x-y+)5-(7x-y+)5=-7x+c.2dy22=(x+)1+4(y
7、+)1+8xy+1dxˊdy222㾷˖ᮍ࣪Ў=x+2x+1+16y+8y+1+8xy+1=(x+4y+)1+2dxdydu1du29Ҹ1+x+4y=uˈ߭݇Ѣx∖ᇐᕫ1+4=ˈ᠔ҹ=u+ˈdxdx4dx41228ߚ⾏ব䞣du=dxˈϸ䖍⿃ߚᕫarctg(+x+y)=6x+cˈᰃdyy6-2x224u+9333=522ॳᮍⱘ㾷DŽdx2xy+xyˊ3223322dy(y)-2xdy3[(y)-2x]3===ˈˈҸy=uˈ߭ॳᮍ࣪Ў23232dxy2(xy+xdx2xy+x㾷˖22ududz3z-6dzdzz
8、-z-6=zˈ߭=z+xˈ᠔ҹ=z+xˈˈx=ˈ..........)1.(xdxdx2z+1dxdx2z+122333uᔧz-z-6=0ˈᕫz=3z=-2ᰃ˄1˅ᮍⱘ㾷DŽेy=3xy=-2xᰃᮍⱘ㾷DŽdu3u2-6x22-622z+11735xᔧz-z-6¹0ᯊˈব䞣ߚ⾏2dz=dxˈϸ䖍⿃ߚ
