微分方程作图市公开课获奖课件省名师优质课赛课一等奖课件.ppt

四川大学数学学院,徐小湛,May 2012,微分方程作图,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,微分方程作图,蜀南竹海 海中海.5.1,1/29,1,with(DEtools):,调用微分方程工具,with(plots):,调用绘图工具,de:=diff(y(x),x)=2*x*y(x);,定义微分方程,fxc:=DEplot(wffc,y(x),x=-2.2,y=-2.2):,画斜率场,jfq:=contourplot(y/exp(x2),x=-2.2,y=-2.2):,画积分曲线,display(fxc,jfqx);,用数学软件Maple能够画出微分方程积分曲线和方向场图形。,画图命令以下：,2/29,2,几何意义：方向场（斜率场）,比如，,微分方程,表示：,曲线,y,=,f,(,x,)在点(,x,y,)处切线斜率为 2,xy,3/29,3,with(DEtools):with(plots):,wffc:=diff(y(x),x)=2*x*y(x):dsolve(wffc);,fangxiangcang:=DEplot(wffc,y(x),x=-2.2,y=-2.2,thickness=2):,jifenquxian:=contourplot(y/exp(x2),x=-2.2,y=-2.2,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,方向场与积分曲线,4/29,4,with(DEtools):with(plots):,wffc:=diff(y(x),x)=(cos(y(x)-y(x)*cos(x)/(x*sin(y(x)+sin(x)-1):,dsolve(wffc);fangxiangcang:=DEplot(wffc,y(x),x=0.4*Pi,y=0.4*Pi,thickness=2):,jifenquxian:=contourplot(y*sin(x)-x*cos(y)-y,x=0.4*Pi,y=0.4*Pi,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,方向场与积分曲线,5/29,5,with(DEtools):with(plots):,wffc:=diff(y(x),x)=(cos(y(x)-y(x)*cos(x)/(x*sin(y(x)+sin(x)-1):,dsolve(wffc);fangxiangcang:=DEplot(wffc,y(x),x=-2.2,y=-2.2,thickness=2):,jifenquxian:=contourplot(y*sin(x)-x*cos(y)-y,x=-2.2,y=-2.2,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,方向场与积分曲线,6/29,6,微分方程：,标准形式:,作出微分方程积分曲线图形。,7/29,7,with(DEtools):with(plots):,wffc:=x*diff(y(x),x)+y(x)=sin(x):dsolve(wffc);,fangxiangcang:=DEplot(wffc,y(x),x=-2.2,y=-2.2,thickness=2):,jifenquxian:=contourplot(x*y+cos(x),x=-2.2,y=-2.2,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,8/29,8,微分方程：,作出微分方程积分曲线图形。,原方程化为：,9/29,9,with(DEtools):with(plots):,wffc:=(x-y(x)3)*diff(y(x),x)+y(x)=0:dsolve(wffc);,fangxiangcang:=DEplot(wffc,y(x),x=-2.2,y=-2.2,thickness=2):,jifenquxian:=contourplot(x*y-y4/4,x=-2.2,y=-2.2,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,10/29,10,with(DEtools):,DEplot(x-y(x)3)*diff(y(x),x)+y(x)=0,y(x),x=-2.2,y=-2.2,y(0)=1,y(0)=0.3,y(0)=1.5,y(0)=-0.5,y(0)=-1,y(0)=-1.5,linecolor=blue,black,gold,navy,green,maroon,color=violet,stepsize=0.01,scaling=constrained);,11/29,11,with(DEtools):with(plots):,wffc:=diff(y(x),x)=2*x*y(x):dsolve(wffc);,fangxiangcang:=DEplot(wffc,y(x),x=-2.2,y=-2.2,thickness=2):,jifenquxian:=contourplot(y/exp(x2),x=-2.2,y=-2.2,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,方向场与积分曲线,12/29,12,with(DEtools):,DEplot(diff(y(x),x)=2*x*y(x),y(x),x=-2.2,y(0)=-1,y(0)=-0.5,y(0)=0,y(0)=0.5,y(0)=1,y(0)=1.5,y=-4.4,linecolor=gold,black,blue,red,brown,green,color=grey,stepsize=0.01,scaling=constrained);,方向场与积分曲线,13/29,13,wffc:=3*x*y(x)2*diff(y(x),x)=x3+y(x)3:dsolve(wffc,implicit);,fangxiangcang:=DEplot(wffc,y(x),x=-2.2,y=-2.2,thickness=2):,jifenquxian:=contourplot(x2-2*y3/x,x=-2.2,y=-2.2,contours=20,color=blue,thickness=2):,display(fangxiangcang,jifenquxian);,方向场及积分曲线,14/29,14,方程通解：,wffc:=diff(y(x),x$2)-diff(y(x),x)-2*y(x)=0:,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-1.1),_C2=-1.1):,plot(toplot,x=-1.1,y=-10.10,thickness=3,color=red);,通解中部分曲线,15/29,15,wffc:=diff(y(x),x$2)-diff(y(x),x)-2*y(x)=0:,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-2.2),_C2=-2.2):,plot(toplot,x=-1.1,y=-10.10,thickness=3,color=red);,更多曲线,方程通解：,16/29,16,求特解：,方程通解：,特解:,17/29,17,通解中部分曲线,和特解曲线,wffc:=diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0:,tongjie:=dsolve(wffc,y(x):tejie:=dsolve(wffc,y(0)=4,D(y)(0)=-2,y(x):,tongjiequxian:=seq(seq(rhs(tongjie),_C1=3.5),_C2=0.4):,p1:=plot(tongjiequxian,x=-2.2,y=0.8,thickness=1,color=red):,p2:=plot(rhs(tejie),x=-2.2,y=0.8,thickness=5,color=blue):,display(p1,p2,scaling=constrained);,18/29,18,wffc:=diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0:,tongjie:=dsolve(wffc,y(x):tejie:=dsolve(wffc,y(0)=4,D(y)(0)=-2,y(x):,tongjiequxian:=seq(seq(rhs(tongjie),_C1=2.6),_C2=0.8):,p1:=plot(tongjiequxian,x=-2.2,y=0.8,color=blue):,p2:=plot(rhs(tejie),x=-2.2,y=0.8,thickness=5,color=red):,display(p1,p2);,19/29,19,方程通解：,20/29,20,通解中部分曲线,wffc:=diff(y(x),x$2)-4*diff(y(x),x)+13*y(x)=0:,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-1.1),_C2=-1.1):,plot(toplot,x=-1.1,y=-10.10,thickness=3,color=red);,21/29,21,通解中部分曲线,wffc:=diff(y(x),x$2)-4*diff(y(x),x)+13*y(x)=0:,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-2.2),_C2=-2.2):,plot(toplot,x=-2.1.5,y=-20.20,thickness=3,color=red);,22/29,22,wffc:=diff(y(x),x$2)-4*diff(y(x),x)+13*y(x)=0:,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-3.3),_C2=-3.3):,plot(toplot,x=-2.1,y=-20.20,thickness=2,color=blue);,23/29,23,方程：,方程通解：,24/29,24,wffc:=diff(y(x),x$2)+y(x)=2*x2-3:,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-3.3),_C2=-3.3):,plot(toplot,x=-2.2,y=-12.6,thickness=2,color=blue);,25/29,25,方程：,通解：,26/29,26,wffc:=diff(y(x),x$2)-2*diff(y(x),x)-3*y(x)=exp(-x):,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-3.3),_C2=-3.3):,plot(toplot,x=-2.2,y=-12.12,color=blue,thickness=2,);,27/29,27,方程：,通解：,28/29,28,wffc:=diff(y(x),x$2)-2*diff(y(x),x)+y(x)=(1+x)*exp(x):,tongjie:=dsolve(wffc,y(x):,toplot:=seq(seq(rhs(tongjie),_C1=-3.3),_C2=-3.3):,plot(toplot,x=-4.3,y=-10.10,color=blue);,29/29,29,