1、微分方程作图 1,微分方程作图,蜀南竹海 海中海 2019.5.1,微分方程作图 2,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可以画出微分方程的的积分曲线和方向场的图形。画图的命令如下：,微分方程作图 3,的几何意义：方向场（斜率场）,例如，,微分方程,表示：,
2、曲线 y=f(x)在点(x,y)处的切线斜率为 2xy,微分方程作图 4,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,微分方程作图
3、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=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)
4、;,方向场与积分曲线,微分方程作图 6,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,ji
5、fenquxian);,方向场与积分曲线,微分方程作图 7,微分方程：,标准形式:,作出微分方程的积分曲线的图形。,微分方程作图 8,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(fangxian
6、gcang,jifenquxian);,微分方程作图 9,微分方程：,作出微分方程的积分曲线的图形。,原方程化为：,微分方程作图 10,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(fangxi
7、angcang,jifenquxian);,微分方程作图 11,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);,微分方程作图 12,with(DEtools):with(plots):wffc:=diff(y(x),x
8、)=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,微分方程作图 13,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)
9、=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,微分方程作图 14,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=-
10、2.2,y=-2.2,contours=20,color=blue,thickness=2):display(fangxiangcang,jifenquxian);,的方向场及积分曲线,微分方程作图 15,方程的通解：,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);,通解中的部分曲线,微分方程作图 16,w
11、ffc:=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);,更多的曲线,方程的通解：,微分方程作图 17,求特解：,方程的通解：,特解:,微分方程作图 18,通解中的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)
12、: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);,微分方程作图 19,wffc:=diff(y(x),x$2)+2*diff(y(x),x)+y(x)=
13、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);,微分方程作图 20,方程的通解：,微分方程作图 21,通解中的部分曲线,wffc:=diff(y(x),x$2)
14、-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);,微分方程作图 22,通解中的部分曲线,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(to
15、plot,x=-2.1.5,y=-20.20,thickness=3,color=red);,微分方程作图 23,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);,微分方程作图 24,方程：,方程的通解：,微分方程作图 25,wffc:=diff(y(x),x$2)+y(x)=2*x2-3:tongj
16、ie:=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);,微分方程作图 26,方程：,通解：,微分方程作图 27,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);,微分方程作图 28,方程：,通解：,微分方程作图 29,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);,
