       Re: Varying a constant in an ODE to Manipulate solution

• To: mathgroup at smc.vnet.net
• Subject: [mg126907] Re: Varying a constant in an ODE to Manipulate solution
• From: Narasimham <mathma18 at hotmail.com>
• Date: Sun, 17 Jun 2012 03:56:16 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201206130857.EAA03730@smc.vnet.net> <jreout\$nqu\$1@smc.vnet.net>

```On Jun 15, 12:41 pm, Murray Eisenberg <mur... at math.umass.edu> wrote:

> It's not clear to me what you want to do.

> Do you want to create 3D-parametric plot of the trajectory in space --

> possibly as a dynamic with the time t as dynamic parameter?

Yes. I can get this plot OK simply as:

Manipulate[ParametricPlot3D[Evaluate[sol[c][]], {t, -3, 3},
PlotStyle ->  {Red, Thick},
AspectRatio ->  Automatic, PlotRange ->  {{-3, 3}, {-10, 10}}], {c,
-0.5, 2, 0.2}]

Also I have asked in a separate but related question now under
clearance by moderators
regarding Clairaut's Equation_

How to include traces of these lines in the plot and not have them
flying in the 3space
on the Manipulate command. Call them t parameter lines as t is
continuous and c is discrete.

> And/or do one of those same things but do it just in 2D, selecting some

> pair of the 3 components of the solution function?

Yes, that was my query. Bob Hanlon pointed out my error. (Evaluating
together was needed).

> Or a 3D-parametric plot of the trajectory over a fixed-duration time in=
terval, but dynamic

> with c as control variable?

Yes, in fact that is going to be my next question now, and thanks that
you have foreseen it !

Call these lines c parameter lines as c would be the continuous
control variable and t's would be discrete.

In this case, each time dot is extruded or curvilinearly dragged
across the surface, cutting the

t lines. Bringing together action of control variables t and c, we
have a surface sol[c_,t_] which

can be computed and plotted. t and c are two parameters for this
patch.

How can this be done? I am almost sure this is doable. I shall try for
Clairaut's 2D problem and send result when done.

Regards

Narasimham

> On 6/13/12 4:57 AM, Narasimham wrote:

> > Same topic is continued. Thanks to Murray Eisenberg and Bob Hanlon
>
> > All variables or a single variable are easily pocked out from sol[c_]
> > list for plotting.
> > But how to pick out two out of them for ParametricPlot  ( 2D) ?
>
> > sol[c_] := {x[t], y[t], z[t]} /.
> >    First[NDSolve[{y''[t] + Sin[y[t]/c] == 0, y' == 0,
> >       y == 1/(1 + c), x'[t] == t, x == c^2,
> >       z'[t] == 2 c x[t] - y[t], z == 2}, {x, y, z}, {t=
, -3, 3}]]
> > Manipulate[
> >   Plot[Evaluate[sol[c]], {t, -3, 3}, PlotStyle ->  {Red, Thick},
> >    AspectRatio ->  Automatic,
> >    PlotRange ->  {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}]
> > Manipulate[
> >   Plot[Evaluate[sol[c][]], {t, -3, 3}, PlotStyle ->  {Red, Thick=
},
> >    AspectRatio ->  Automatic,
> >    PlotRange ->  {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}]
> > " 2 parameter Dynamic manipulation not OK"
> > Manipulate[
> >   Plot[{Evaluate[sol[c][]], Evaluate[sol[c][]]}, {t, -3, 3},
> >    PlotStyle ->  {Green, Thick}, AspectRatio ->  Automatic,
> >    PlotRange ->  {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}]
>
> --
> Murray Eisenberg                     murrayeisenb...@=
gmail.com
> 80 Fearing Street                    phone 413 549-10=
20 (H)
> Amherst, MA 01002-1912

```

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