微分方程作图

微分方程作图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(x^2),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：2的几何意义：方向场（斜率场）例如，微分方程表示：曲线y=f(x)在点(x,y)处的切线斜率为2xy3with(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(x^2),x=-2..2,y=-2..2,contours=20,color=blue,thickness=2):display(fangxiangcang,jifenquxian);方向场与积分曲线4with(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5with(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微分方程：标准形式:作出微分方程的积分曲线的图形。7with(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微分方程：作出微分方程的积分曲线的图形。原方程化为：9with(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-y^4/4,x=-2..2,y=-2..2,contours=20,color=blue,thickness=2):display(fangxiangcang,jifenquxian);10with(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);11with(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(x^2),x=-2..2,y=-2..2,contours=20,color=blue,thickness=2):display(fangxiangcang,jifenquxian);方向场与积分曲线12with(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13wffc:=3*x*y(x)^2*diff(y(x),x)=x^3+y(x)^3:dsolve(wffc,implicit);fangxiangcang:=DEplot(wffc,y(x),x=-2..2,y=-2..2,thickness=2):jifenquxian:=contourplot(x^2-2*y^3/x,x=-2..2,y=-2..2,contours=20,color=blue,thickness=2):display(fangxiangcang,jifenquxian);的方向场及积分曲线14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15wffc:=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：16求特解：方程的通解：特解:17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)):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);18wffc:=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方程的通解：20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=-1..1),_C2=-1..1)]:plot(toplot,x=-1..1,y=-10..10,thickness=3,color=red);21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(toplot,x=-2..1.5,y=-20..20,thickness=3,color=red);22wffc:=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方程：方程的通解：24wffc:=diff(y(x),x$2)+y(x)=2*x^2-3: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方程：通解：26wffc:=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)]:p