> with(DEtools):
> DE:=diff(y(t),t)=4*t/y(t);
> DEplot(DE,y(t),t=-2..2,[[y(0)=0.1],[y(1)=-.5]],y=-2..2);
> dsolve(DE,y(t));
> DE1:=diff(y(t),t)=1-y(t);
> DEplot(DE1,y(t),t=-5..5,y=-3..3);
> DEplot(DE1,y(t),t=-5..5,[[y(0)=-2],[y(1)=1]],y=-3..3);
> dsolve(DE1,y(t));
> DE2:=diff(y(t),t)=t+y(t);
> DEplot(DE2,y(t),t=-5..5,[[y(0)=-1],[y(0)=-3]],y=-5..5);
>
> dsolve(DE2,y(t));
> DE3:=diff(y(t),t)=2*t*(y(t))^2;
> DEplot(DE3,y(t),t=-5..5,[[y(0)=1],[y(0)=-3]],y=-4..4);
> dsolve(DE3,y(t));
>
dfieldplot([diff(x(t),t)=x(t)+y(t), diff(y(t),t)=x(t)-3*y(t)],
[x(t),y(t)],t=-2..2, x=-2..2, y=-2..2, arrows=LARGE,
title=`Linear DE`, color=[x(t)+y(t), x(t)-3*y(t),.1]);
>
dsolve([diff(x(t),t)=x(t)+y(t), diff(y(t),t)=x(t)-3*y(t)],
[x(t),y(t)]);
>
dfieldplot([diff(R(t),t)=R(t)-R(t)*F(t), diff(F(t),t)=-2*F(t)+2*R(t)*F(t)],
[R(t),F(t)],t=-2..2, R=-0..2, F=-0..2, arrows=LARGE,
title=`Preditor-prey model`, color=[R(t)-R(t)*F(t), -2*F(t)+2*R(t)*F(t),.1]);
>
dsolve([diff(R(t),t)=R(t)-R(t)*F(t), diff(F(t),t)=-2*F(t)+2*R(t)*F(t)],
[R(t),F(t)]);
>
dfieldplot([diff(y(t),t)=v(t), diff(v(t),t)=-2*y(t)-2*v(t)],
[y(t),v(t)],t=-2..2, y=-2..2, v=-2..2, arrows=LARGE,
title=`underdamped harmonic oscillator`, color=[v(t), -2*y(t)-2*v(t),.1]);
>