How does your response give us any new information to better help you?

What have you tried? Be explicit, show us; upload a Maple worksheet.

Can you respond to any of the questions in my first response?

Simply telling me you don't know where to start is not a question I can answer. I have already made several assumptions and presented several alternatives in a first attempt to help.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

How does your response give us any new information to better help you?

What have you tried? Be explicit, show us; upload a Maple worksheet.

Can you respond to any of the questions in my first response?

Simply telling me you don't know where to start is not a question I can answer. I have already made several assumptions and presented several alternatives in a first attempt to help.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

acer's response is more than I would offer.

You do not need to know the Maple command to get useful information from the help system. Try, for example,

?derivative
?limit
?linear algebra

That's as much help as I would give, leaving the student to determine the best tools to use in each case. I would also add a comment that you should always look at the See Also section at the bottom of the help page.

Lastly, I would also suggest trying a Google search for say "maple derivative".

Even in today's economy, that's more than my two cents worth of input,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

acer's response is more than I would offer.

You do not need to know the Maple command to get useful information from the help system. Try, for example,

?derivative
?limit
?linear algebra

That's as much help as I would give, leaving the student to determine the best tools to use in each case. I would also add a comment that you should always look at the See Also section at the bottom of the help page.

Lastly, I would also suggest trying a Google search for say "maple derivative".

Even in today's economy, that's more than my two cents worth of input,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

The original request is out of line. It exaggarates the real situation by a factor of more than 70,000,000. There are only 13 problems on the test. These questions appear to be practice problems for an exam.

For the student's sake, I just hope the professor is not a regular visitor to MaplePrimes.

The questions are very reasonable, but I can't believe the students have had no prior experience with Maple. At the least, the student could show the initial efforts - if there are any initial efforts. I have to admit that I'm skeptical that the student spent any time on this except to type the original message in this thread.

In short:  NO DEAL!

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

The original request is out of line. It exaggarates the real situation by a factor of more than 70,000,000. There are only 13 problems on the test. These questions appear to be practice problems for an exam.

For the student's sake, I just hope the professor is not a regular visitor to MaplePrimes.

The questions are very reasonable, but I can't believe the students have had no prior experience with Maple. At the least, the student could show the initial efforts - if there are any initial efforts. I have to admit that I'm skeptical that the student spent any time on this except to type the original message in this thread.

In short:  NO DEAL!

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Maple 8 was created long before Vista was released. This version of Maple is not compatible with Vista. Maple 11 is the first release of Maple that is supported under Vista.

For more info, click here.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

To get everything in this one post, let's recall the setting from my initial response:

ODE := diff(y(t),t$2)+y(t)=piecewise(t in [0,Pi],t, t>Pi,Pi);
ICS := y(0)=0, D(y)(0)=0;
ODE2 := diff(y(t),t$2)+y(t)=piecewise(t <Pi,t, t>Pi,Pi);

To precisely define the RHS as you were trying to do in your original post I would use:

ODE3 := diff(y(t),t$2)+y(t)=piecewise( t>=0 and t <=Pi, t , t>Pi, Pi ):

But, this has no affect on the solution of the IVP for t>=0:

dsolve( {ODE3,ICS}, y(t) );
  y(t) = piecewise(t < 0, 0, t < Pi, -sin(t) + t, Pi <= t, -2 sin(t) + Pi)

To compare the three different right-hand sides of the ODEs, see

plot( map(rhs,[ODE,ODE2,ODE3]), t=-1..5,
      style=point, symbol=[asterisk,cross,circle,asterisk], symbolsize=15,
      color=[green,red,blue], numpoints=50, adaptive=false );

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

To get everything in this one post, let's recall the setting from my initial response:

ODE := diff(y(t),t$2)+y(t)=piecewise(t in [0,Pi],t, t>Pi,Pi);
ICS := y(0)=0, D(y)(0)=0;
ODE2 := diff(y(t),t$2)+y(t)=piecewise(t <Pi,t, t>Pi,Pi);

To precisely define the RHS as you were trying to do in your original post I would use:

ODE3 := diff(y(t),t$2)+y(t)=piecewise( t>=0 and t <=Pi, t , t>Pi, Pi ):

But, this has no affect on the solution of the IVP for t>=0:

dsolve( {ODE3,ICS}, y(t) );
  y(t) = piecewise(t < 0, 0, t < Pi, -sin(t) + t, Pi <= t, -2 sin(t) + Pi)

To compare the three different right-hand sides of the ODEs, see

plot( map(rhs,[ODE,ODE2,ODE3]), t=-1..5,
      style=point, symbol=[asterisk,cross,circle,asterisk], symbolsize=15,
      color=[green,red,blue], numpoints=50, adaptive=false );

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

The phase portrait arrow routines I wrote last night for the discussion in Phase diagram for k(dot) and k can also be used in this analysis.

We start by putting all of the definitions from Robert's post in one place:

restart: with(plots): with(DEtools):
a:= 0.04: b:= 0.06: c:= (a+b)/2:
f:= x-> 20*x^(0.5):
g:= x-> 10*x^(-0.5):
zss:=b: zss: xss:=fsolve(g(x)-zss,x): xss: yss:=f(xss): yss:
xdot:= diff(x(t),t) = f(x(t))-y(t):
ydot:= diff(y(t),t) = (g(x(t))-z(t))*y(t):
zdot:= diff(z(t),t) = (z(t)-a)*(c-z(t))+(z(t)-b)*(g(x(t))-z(t))*y(t)/(f(x(t))-y(t)):
xd:=eval(rhs(xdot),{x(t)=x, y(t)=y, z(t)=z}):
yd:=eval(rhs(ydot),{x(t)=x, y(t)=y, z(t)=z}):
zd:=eval(rhs(zdot),{x(t)=x, y(t)=y, z(t)=z}):
zss:=b: zss; xss:=fsolve(g(x)-zss,x): xss; yss:=f(xss): yss;
opts:=fieldstrength=average(0.5), axes=boxed, grid=[5,5,5], arrows=SLIM, thickness=2, tickmarks=[4,4,4], orientation=[-15,60]:
SYS :={xdot, ydot, zdot}:
VAR := [x(t),y(t),z(t)]:
INI := [[x(0)=0.99*xss, y(0)=1.01*yss, z(0)=0.99*zss], [x(0)=1.01*xss, y(0)=1.01*yss, z(0)=0.99*zss], [x(0)=0.99*xss, y(0)=1.01*yss, z(0)=1.01*zss], [x(0)=1.01*xss, y(0)=1.01*yss, z(0)=1.01*zss]]:

sys:=convert([xdot,ydot,zdot],rational);
PDEtools[dchange]({x(t)=u(t)^2,y(t)=w(t)+20*u(t),z(t)=v(t)+10/u(t)},       sys,[u(t),v(t),w(t)]);
simplify(%) assuming u(t)>0;
newsys:= expand(solve(%,        {diff(u(t),t), diff(v(t), t), diff(w(t),t)})) ;

eval(newsys, u(t) = 100);
sys100:= select(has,%, diff);

vp := subs(sys100, v(t)=v, w(t)=w, diff(v(t),t));
wp := subs(sys100, v(t)=v, w(t)=w, diff(w(t),t));

Here are the two routines for creating arrows in the phase portrait

PParrow2 := proc( sys, pt )
  local opt;
  if  _npassed>2 then opt := _passed[3..-1] else opt := NULL end if;
  plots:-display( [ PParrow1(sys[1],pt,<1,0>,opt),
                    PParrow1(sys[2],pt,<0,1>,opt) ] )
end proc:
PParrow1 := proc( eqn, pt, dir )
  local d, p, opt;
  if  _npassed>3 then opt := _passed[4..-1] else opt := NULL end if;
  d := signum( eval(eqn,pt) )*dir;
  p  := map(rhs,pt);
  plots:-arrow(p,d,opt)
end proc:

And here is how the result that I obtain, showing the nullclines and the flow of solutions in each region defined by the nullclines:

display( [
  implicitplot([vp,wp], w=-0.1  .. 0.1, v = -0.001 .. 0.001, colour=[red,blue],gridrefine=3),
  PParrow1(vp,[w= 0.05,v= 1e-6],<0,1>,length=0.25e-6,width=0.5e-3),
  PParrow1(wp,[w= 0.05,v= 1e-6],<1,0>,length=0.5e-2,width=1e-8),
  PParrow1(vp,[w= 0.05,v= 2e-6],<0,1>,length=0.25e-6,width=0.5e-3),
  PParrow1(wp,[w= 0.05,v= 2e-6],<1,0>,length=0.5e-2,width=1e-8),
  PParrow1(vp,[w= 0.05,v= 3e-6],<0,1>,length=0.25e-6,width=0.5e-3),
  PParrow1(wp,[w= 0.05,v= 3e-6],<1,0>,length=0.5e-2,width=1e-8),
  PParrow1(vp,[w=-0.05,v=-1e-6],<0,1>,length=0.25e-6,width=0.5e-3),
  PParrow1(wp,[w=-0.05,v=-1e-6],<1,0>,length=0.5e-2,width=5e-9,head_width=5e-8),
  PParrow1(vp,[w=-0.05,v=-2.3e-6],<0,1>,length=0.25e-6,width=0.5e-3),
  PParrow1(wp,[w=-0.05,v=-2.3e-6],<1,0>,length=0.5e-2,width=1e-10,head_width=1e-9),
  PParrow1(vp,[w=-0.05,v=-3e-6],<0,1>,length=0.25e-6,width=0.5e-3),
  PParrow1(wp,[w=-0.05,v=-3e-6],<1,0>,length=0.5e-2,width=5e-9,head_width=5e-8),
  NULL ] );

This is consistent with the previous plots, but with a little additional information. Note the use of options from plots,arrow to deal with the different scales on the axes.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Robert,

Can  you post your worksheet? I have had trouble getting any plots for this problem. I'm guessing that you might be doing something slightly different. What I want to investigate is the horizontal movement of solutions. I believe you have accurately reported the vertical nature of solutions.

The results you have reported scream out to me for nondimensionalization. I have not tried to find appropriate scales for dependent or independent variables. My guess is that this will be very enlightening.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Look carefully, at what you typed.

I do not know how to get x-dot in a plot label. You might be able to do this with the Typesetting package. Personally, I think I would just go with one of  "x ' ", D(x), or diff(x(t),t).

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Look carefully, at what you typed.

I do not know how to get x-dot in a plot label. You might be able to do this with the Typesetting package. Personally, I think I would just go with one of  "x ' ", D(x), or diff(x(t),t).

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Where did  you put the dirgrid? It is an argument to DEplot, so it has to go in the second argument to the animate command. If you put dirgrid as an argument to animate, it won't know what to do (and will ignore it). Here's how I would write it:

animate( DEplot,
           [ SYS, [x(t),y(t)], t=0..10, x=-3..3,
             y=-3..3, scene=[x(t),y(t)],
             [[x(0)=1,y(0)=0],[x(0)=2,y(0)=0]],
             dirgrid=[80,80] ],
         epsilon=0..1, frames=11 );

If you are trying to get more solution curves, then you will have to add additional initial conditions to the list of two that are currently given.

Does this resolve the problem you are having?

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Where did  you put the dirgrid? It is an argument to DEplot, so it has to go in the second argument to the animate command. If you put dirgrid as an argument to animate, it won't know what to do (and will ignore it). Here's how I would write it:

animate( DEplot,
           [ SYS, [x(t),y(t)], t=0..10, x=-3..3,
             y=-3..3, scene=[x(t),y(t)],
             [[x(0)=1,y(0)=0],[x(0)=2,y(0)=0]],
             dirgrid=[80,80] ],
         epsilon=0..1, frames=11 );

If you are trying to get more solution curves, then you will have to add additional initial conditions to the list of two that are currently given.

Does this resolve the problem you are having?

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed