phug:=[diff(theta(t),t)=(v(t)^2-cos(theta(t)))/v(t),diff(v(t),t)=-sin(theta(t))-R*v(t)^2];R:=0.1;with(DEtools):We did last time this graphic of the phug with standard methods.DEplot(phug,[theta(t),v(t)],t=0..5,[seq([theta(0)=0,v(0)=1+0.2*i],i=0..10)],theta=-Pi..3*Pi,v=0.01..3,
linecolor=black,stepsize=0.1);DEplot(phug,[theta(t),v(t)],t=0..5,[seq([theta(0)=0,v(0)=1+0.2*i],i=0..10)],theta=-Pi..3*Pi,v=0.01..3,linecolor=black,method=classical[foreuler]);DEplot(phug,[theta(t),v(t)],t=0..5,[seq([theta(0)=0,v(0)=1+0.2*i],i=0..10)],theta=-Pi..3*Pi,v=0.01..3,linecolor=black,method=classical[heunform]);R:='R';xphug:=[ diff(theta(t),t) =(v(t)^2-cos(theta(t)))/v(t),
diff(v(t),t) =-sin(theta(t))-R*v(t)^2,
diff(x(t),t) =v(t)*cos(theta(t)),
diff(y(t),t) =v(t)*sin(theta(t))];R:=0;with(plots):R:=0;
display(array([
DEplot(xphug,[theta,v,x,y],t=0..15,[[theta(0)=0,v(0)=1.5,x(0)=0,y(0)=1],[theta(0)=0,v(0)=2.5,x(0)=0,y(0)=2]],
theta=-Pi..5*Pi/2,v=0.1..3,x=-1..10,y=0..4,linecolor=black,stepsize=0.1,scene=[theta,v]),
DEplot(xphug,[theta,v,x,y],t=0..15,[[theta(0)=0,v(0)=1.5,x(0)=0,y(0)=1],[theta(0)=0,v(0)=2.5,x(0)=0,y(0)=2]],
theta=-Pi..5*Pi/2,v=0.1..3,x=-1..10,y=0..4,linecolor=blue,stepsize=0.1,scene=[x,y])
]));R:=0.2;
display(array([
DEplot(xphug,[theta,v,x,y],t=0..15,[[theta(0)=0,v(0)=1.5,x(0)=0,y(0)=1],[theta(0)=0,v(0)=2.5,x(0)=0,y(0)=2]],
theta=-Pi..5*Pi/2,v=0.1..3,x=-1..10,y=0..4,linecolor=black,stepsize=0.1,scene=[theta,v]),
DEplot(xphug,[theta,v,x,y],t=0..15,[[theta(0)=0,v(0)=1.5,x(0)=0,y(0)=1],[theta(0)=0,v(0)=2.5,x(0)=0,y(0)=2]],
theta=-Pi..5*Pi/2,v=0.1..3,x=-1..10,y=0..4,linecolor=blue,stepsize=0.1,scene=[x,y])
]));R:=3;
display(array([
DEplot(xphug,[theta,v,x,y],t=0..15,[[theta(0)=0,v(0)=1.5,x(0)=0,y(0)=1],[theta(0)=0,v(0)=2.5,x(0)=0,y(0)=2]],
theta=-Pi..5*Pi/2,v=0.1..3,x=-1..10,y=0..4,linecolor=black,stepsize=0.1,scene=[theta,v]),
DEplot(xphug,[theta,v,x,y],t=0..15,[[theta(0)=0,v(0)=1.5,x(0)=0,y(0)=1],[theta(0)=0,v(0)=2.5,x(0)=0,y(0)=2]],
theta=-Pi..5*Pi/2,v=0.1..3,x=-1..10,y=0..4,linecolor=blue,stepsize=0.1,scene=[x,y])
]));R:='R';
FixPoint:convert(
solve({(v^2-cos(theta))/v=0,-sin(theta)-R*v^2=0},{theta,v}),
radical);sample:=[diff(y(t),t)=d*y(t),diff(x(t),t)=a*x(t)];a:=4;
d:=2;
DEplot(sample,[x(t),y(t)],t=0..5,[seq([x(0)=0.1,y(0)=0.2*i],i=-10..10)],
x=-10..10,y=-10..10,
linecolor=black,stepsize=0.1);a:=4;
d:=2;
DEplot(sample,[x(t),y(t)],t=0..5,[seq([x(0)=-0.1,y(0)=0.2*j],j=-10..10)],
x=-10..10,y=-10..10,linecolor=black,stepsize=0.1);sample2:=[diff(x(t),t)=x(t)+5*y(t), diff(y(t),t)=-x(t)-3*y(t)];DEplot(sample2,[x(t),y(t)],t=-5..5,[seq([x(0)=-0.1,y(0)=0.2*j],j=-10..10)],
x=-5..5,y=-4..4,linecolor=black,stepsize=0.1);