《常微分方程》-(方道元-著)-课后习题答案--浙江大学出版社

1.1000/-20000/2ha=¡2oooom=h2v0=1000m=h,T8:dhdt=v0+at:h(T)=0dhdtjt=T=0h=v0t+12at2+C,Cv0T+12aT2+C=0,T=¡v0ah(0)=C=25m.2.VAV1/AV2/V1+V2/2000A5m02000Am05AAP(t),8:VP0(t)+P(t)(V1+V2)=m05V1P(0)=5m0P(t)=e¡V1+V2Vt[5m0+m05V1V1+V2(eV1+V2Vt¡1)].3.(x;y)v0v18:dxdt=v0dydt=v1x(0)=0;y(0)=04.200kg43¼150kg¼¸ºc¸ºsºcºs¸1 mc;msVc;Vs,8:mcg¡¸Vc=mcdVcdt+½Vcgmsg¡¸Vs=msdVsdt+½Vsg8:mcg¡½Vcg=Mcmsg¡½Vsg=MsMcMs=438:Vc(t)=Mcmc¸(1¡e¡¸mct)Vs(t)=Msms¸(1¡e¡¸mst)mcms=431.(1)dydx=4x3¡ysinx;(2)dydx2¡(dydx)2+2xy;(3)x2d2ydx2¡2xdydx+y=2xsinx;(4)dydx+cosy+4x=0;(5)yd3ydx3¡exdydx+3xy=0;(6)d3ydx3+3dydx¡6y=0:(1)(2)(3)(4)(5)(6)2.(1)y=1+x2;dydx=y2¡(x2+1)y+2x;(2)y=¡1x;x2dydx¡x2y2¡xy¡1=0;(3)y=C1e2x+C2e¡2x;dydx2¡4y=0(C1C2);(4)y=cxex;d2ydx2¡2dydx+y=0(c);(5)y=ecx;(dydx)2¡yd2ydx2=0(c);2 (6)y=8:¡(x¡C1)24;¡1xC1;0;C1xC2;(x¡C1)24;C2x+1;dydx=pjyj:(1)(2)(3)(4)(5)(6)3.(1)y=Cx+x2(2)xy=C(3)(4)y=92C+Cxx2x(1)y00=2(2)y+xy0=0(3)y000+(y0)2y000=3y0(y00)2(4)y000=01.(1)dydx=y+sinx;(2)d2ydx2¡11¡x2y=1+x;y(0)=1;(3)y=ex+Rx0y(t)dt;(4)dydx=x4+y3xy2;(5)2xydy¡(2y2¡x)dx=0;(6)(ylnx¡2)ydx=xdy;(7)3xy2dydx+y3+x3=0;(8)dydx=yx+y3:3 (1)e¡xy=Cex¡sinx+cosx2C(2)e¡R11¡x2dxy=r1+x1¡x(C+¼2)CC=1¡¼2y=q1+x1¡x(3)xy0=ex+y(x)y(0)=1e¡xy=ex(C+x)CC=1y=ex(1+x)(4)3y2z=y3dzdx=3x3+3zxe¡R3xdxz=x3(C+3x)Cy3=x3(C+3x)(5)z=y2dzdx=2zx¡1e¡R2xdxz=x2(C+x¡1y2=x2(C+x¡1)C(6)y´0y6=0¡y¡2z=y¡1dzdx=2zx¡lnxxe¡R2xdxy¡1=x2(C+1+2lnx4x2)4  C(7)y=0y6=0y0=¡13xy¡x23y23y2z=y3eR1xdxy=Cx¡x34Cd(xy3)dx=¡x3(8)y´0y6=0ydx¡xdyy2¡ydy=0xy¡y22=CC2.y='(x)dydx+a(x)y·0;(x¸0):'(x)·'(0)e¡Rx0a(t)dt;(x¸0):eRx0a(s)ds3.f(x)(¡1;+1)dydx¡y=f(x)(¡1;+1)f(x)!!jf(x)j·Me¡xy=ex(C+Rf(s)e¡sds)Cjyj·Cex+M(¡1;+1)C=0y=exRf(s)e¡sdsf(x+!)=f(x)y(x+!)=e(x+!)Rf(s+!)e¡(s+!)ds=e(x+!)e¡!Rf(s+!)e¡sds=y(x)4.dydxp(x)y+q(x);p(x)q(x)!(1)q(x)´0(2.4.23)!p(x)¹p=1!Z!0p(x)dx=05 (2)q(x)6=0(2.4.23)!p(x)¹p6=0y(x)=y(x+!),C=Ce!¹p+e!¹pZ!0q(t)e¡Rt0p(s)dsdt(1)q(t)=0y(x)=y(x+!),¹p=0(2)q(t)6=0y(x)=y(x+!),¹p6=0y=eRp(x)dx(C+Zq(x)e¡Rp(x)dxdx);C=11¡e!¹pZ!0q(t)e¡Rt0p(s)dsdt1.(1)xdydx¡4xy=x2py;(2)dydx¡xy2(x2¡1)¡x2y=0;y(0)=1;(3)dydx=y2+14x2;(4)x2dydx¡x2y2=xy+1;(5)dydx=1¡x+y2¡xy2;(6)dydx=ex+y+3;(7)cosysinxdydx=sinycosx;(8)2xydydx=3y2¡x2;(9)(x¡pxy)dydx=y;(10)exydydx+y(1+exy)=0;(11)2xsiny+y3ex+(x2cosy+3y2ex)dydx=0;(12)y22¡2yex+(y¡ex)dydx=0;(13)1+(1+xy)exy+(1+x2exy)dydx=0;(14)ysec2x+secxtanx+(2y+tanx)dydxdydx=0;(15)ydx¡(x2+y2+x)dy=0;(16)y(1+xy)dx¡xdy=0:(1)y´0y6=012y¡12z=y12e¡2xy=(Ce2x¡x4¡18)26  C(2)2yz=y2e¡Rxx2¡1dxy=p1¡x2(C¡p1¡x2)CCy=2p1¡x2¡(1¡x2)(3)z=xyz0=4z2+4z+14xz=¡12z6=¡12y=1Cx¡xlnjxj¡12xCxy=¡12y=1Cx¡xlnjxj¡12xu=y+12xuu0+ux¡u2=0(4)xy=uu0=u2+2u+1xu2+2u+1=0xy=¡1xy6=¡1lnjxj+1xy+1=Cxy=¡1lnjxj+1xy+1=CC(5)y=tan(x¡12x2+C)C(6)e¡y+ex+3=CC(7)siny=0y=k¼(k2Z)siny6=0sin2y¡Csin2x=0C(8)z=y2z0=3xz¡xe¡R3xdxy2=jxj3(C+jxj¡1)C(9)u=yxux2(1¡pu)u0=xupuu=0y=0u6=0y=Ce¡2pxyC(10)u=xyydudy=¡u+eu1+eux+yexy=CC(11)[(x2cosy)dy+(2xsiny)dx]+[(y3ex)dx+(3y2ex)dy]=07  x2siny+y3ex=CC(12)[y22ex¡2ye2x]dx+(exy¡e2x)dy=0y22ex¡ye2x=CC(13)dy+dx+fx2exydy+[(1+xy)exy]dxg=0y+x+xexy=CC(14)(ysec2x)dx+tanxdy+(secxtanx)dx+2ydy=0ytanx+secx+y2=CC(15)y=0y6=0xyd(lnxy)=(yx+xy)dyz=lnxyarctanxy¡y=CC(16)y=0y6=01y21ydx¡xy2dy+xdx=0x+12x2y=CyC2.dydx=x+y+1x¡y+38  13x=X+h;y=Y+k(a)hkdydx=x+y+1x¡y+3dYdX=X+YX¡Y(b)dydx=x+y+1x¡y+3(1)x=X+h;y=Y+kdYdX=X+YX¡Y8:h+k+1=0h¡k+3=0h=¡2;k=1(2)arctanu¡lnp1+u2=lnjxj+CCu=y¡1x+23.dydx=ax+by+mcx+dy+nabcdmnad¡bc6=0dYdX=aX+bYcX+dYad=bcad¡bc6=0ax+by+m=0cx+dy+n=0(nb¡mdad¡bc;an¡mcbc¡ad)Y=y¡an¡mcbc¡ad;X=x¡nb¡mdad¡bcdYdX=aX+bYcX+dYad=bc¹¹(ax+by+m)=(cx+dy+n)dydx=¹y+¹x+CC¹;¹(ax+by+m)6=(cx+dy+n)a;b;c;da6=0c=0d=0dydx=ax+by+mn8:¡abxe¡bnx¡anb2e¡bnx¡ye¡bnx¡mbe¡bnx=Cb6=0y=a2nx2+mnx+Cb=09 c6=0z=ax+by+ancdzdx=(abc+a)+ab(m¡anc)cz:=A+BzzA¡BA2lnjA(Az+B)j=x+C4.a(a)x+ye2xy+axe2xydydx=0(b)1x2+1y2+ax+1y3dydx=0(a)a=112x2+12e2xy+C=0(b)a=¡2xy2¡1x¡12y2+C=05.M(t)+N(y)dydt=0@M(t)@y=0=@N(y)@t6.Bernoulli(1)dydx=f(yx)u=yx(f(u)¡u)dx¡xdu=0M=f(u)¡u;N=¡x@N@x¡@M@uM=¡f0(u)f(u)¡ueR¡f0(u)f(u)¡udu(2)Bernoullidydx+g(x)y+h(x)y®=0(®6=0;1)z=y¡®+1dz+[(1¡®)g(x)z+(1¡®)h(x)]dx=0M=[(1¡®)g(x)z+(1¡®)h(x)];N=1@N@z¡@M@xM=(1¡®)g(x)e(1¡®)Rg(x)dx7.P(x;y)dx+Q(x;y)dy=0¹(x+y)¹(xy)¹1¹(Q@¹@x¡P@¹@y)=@P@y¡@Q@x!=x+y(Q¡P)d¹d!=(@P@y¡@Q@x)¹@P@!¡@Q@!=0¹(x+y)@P@!¡@Q@!6=0(1)1Q¡P(@P@y¡@Q@x))x+y¹(xy)1Qy¡Px(@P@y¡@Q@x)xy1.(1)x=yy0+a(y0)2;(2)16x2+2y(y0)2¡x(y0)3=0;(3)y=2xy0+x2(y0)4;(4)(y0)2cos2y+y0sinxcosxcosy¡sinycos2x=0;(5)(xy0¡y)(yy0+x)=2y0;(6)9(y0)2+4y2=1:10  p=dydx(1)x=yp+ap2x(yp+2ap2)dp+(p2¡1)dy=0p=§1y=§x+ap26=11pp2¡18:ypp2¡1+a[ppp2¡1+lnjp+pp2¡1j]=Cjpj1y(¡p1¡p2)+a[arcsinp¡pp1¡p2]=Cjpj1Cp-(2)y=xp2¡8x2p2x(p2¡16xp2)(1¡xpp0)=0y=¡3£2¡13x432C2yx2=C3x4¡16x2CC-(3)y=2xp+x2p4x(p+2xp0)(1+2xp3)=0y=¡3£2¡43x23(y¡C4)2=4C2jxjCC-(4)u=siny;v=sinx(u0)2+u0vv0¡u(v0)2=0p=dudv;t=dpdv(2p+v)t=0p=¡v2t=04siny+sin2x=0siny=Csinx+C2CC-(5)2yu=y2;p=u02u=xp¡2px+p2x[(p+2x)2¡8](xp0¡p)=0(p+2x)2=8xp0=pp=¡2x§2p2p=CjxjCp=¡2x§2p2u0u=y2¸0(§Cxjxj¡2y2)(§C2jxj+x)=§2CjxjC(6)y2u=y2;p=u036p2+(8u¡1)2=1x=s;6p=sint;8u¡1=costdu=pdxsint(ds6+dt8)=0sint=0ds6+dt80p=0s+34t=Cy=0y=§12sin2(43x+C)+64y4¡16y2=0CC-2.xyy=y(x)y=y(x)(x;y)O(x;y)dydxyx=¡1x2+y2=CC3.xy2=4c(x+c)dydx¡dxdyxC=12yy0y=2xy0+y(y0)2dydx¡dxdy=¡1y04()11 8:F(x;y)=CG(x;y)=C8