> with(DEtools):

> DE:=diff(y(t),t)=4*t/y(t);

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);

[Maple Plot]

> dsolve(DE,y(t));

y(t) = sqrt(4*t^2+_C1), y(t) = -sqrt(4*t^2+_C1)

> DE1:=diff(y(t),t)=1-y(t);

DE1 := diff(y(t),t) = 1-y(t)

> DEplot(DE1,y(t),t=-5..5,y=-3..3);

[Maple Plot]

> DEplot(DE1,y(t),t=-5..5,[[y(0)=-2],[y(1)=1]],y=-3..3);

[Maple Plot]

> dsolve(DE1,y(t));

y(t) = 1+exp(-t)*_C1

> DE2:=diff(y(t),t)=t+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);

>

[Maple Plot]

> dsolve(DE2,y(t));

y(t) = -t-1+exp(t)*_C1

> DE3:=diff(y(t),t)=2*t*(y(t))^2;

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);

[Maple Plot]

> dsolve(DE3,y(t));

y(t) = 1/(-t^2+_C1)

> 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]);

[Maple Plot]

> dsolve([diff(x(t),t)=x(t)+y(t), diff(y(t),t)=x(t)-3*y(t)],
[x(t),y(t)]);

{x(t) = _C1*exp((sqrt(5)-1)*t)*sqrt(5)+2*_C1*exp((s...
{x(t) = _C1*exp((sqrt(5)-1)*t)*sqrt(5)+2*_C1*exp((s...

> 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]);

[Maple Plot]

> 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)]);

[{F(t) = 0}, {R(t) = _C1*exp(t)}], [{F(t) = RootOf(...
[{F(t) = 0}, {R(t) = _C1*exp(t)}], [{F(t) = RootOf(...
[{F(t) = 0}, {R(t) = _C1*exp(t)}], [{F(t) = RootOf(...

> 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]);

[Maple Plot]

>