08-03-06.mw

 > myeq:=y=3*x+7;

 (1)

 > solve(myeq,{x});

 (2)

 > solve(y=3*x+7,{x});

 (3)

 >

 >

 > with(DEtools):

 > R:='R'; phug:= [ diff(theta(t),t) = ( v(t)^2 - cos(theta(t))) / v(t),         diff(v(t),t)     = -sin(theta(t)) - R*v(t)^2 ];

 (4)

 > R:=0;DEplot( phug, [ theta(t), v(t) ], t=0..6,

 > [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

 > theta=-Pi/2..2*Pi, v=0..3,

 > linecolor=[seq(COLOR(HUE, i/10), i=0..10)]);

 >

 >

 > R:=0;DEplot( phug, [ theta(t), v(t) ], t=0..6,

 > [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

 > theta=-Pi/2..2*Pi, v=0..3,

 > linecolor=[seq(COLOR(HUE, i/10), i=0..10)], method=classical[foreuler],      stepsize=1);

 >

 >

 > DEplot( phug, [ theta(t), v(t) ], t=0..6,

 > [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

 > theta=-Pi/2..2*Pi, v=0..3,

 > linecolor=[seq(COLOR(HUE, i/10), i=0..10)], method=classical[foreuler],        stepsize=.1);

 >

 > DEplot( phug, [ theta(t), v(t) ], t=0..6,

 > [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

 > theta=-Pi/2..2*Pi, v=0..3,

 > linecolor=[seq(COLOR(HUE, i/10), i=0..10)], method=classical[rk4],        stepsize=.1);

 >

 > R:=0.1;

 > DEplot( phug, [ theta(t), v(t) ], t=0..6,

 > [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

 > theta=-Pi/2..4*Pi, v=0..3,

 > linecolor=[seq(COLOR(HUE, i/10), i=0..10)], method=classical[rk4],        stepsize=.1);

 >

 > DEplot( phug, [ theta(t), v(t) ], t=0..20,

 > [[theta(0)=0,v(0)=2]],

 > theta=-Pi/2..8, v=0..3,

 > linecolor=[seq(COLOR(HUE, i/10), i=0..10)], method=classical[rk4],        stepsize=.01);

 >

 >

 >