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Mathematica Calculus Equation Solving Wolfram Language

Plotting DSolve outputs?

Tim Kirkpatrick

Tim Kirkpatrick, Non Affiliated

Posted 11 years ago

I'm creating a template for differential equation solving. I'm trying to write a script that takes the output from the solution of the differential equation and write it as a new function to be used in Manipulate [ Plot[...] ]. However, when I do this, Plot will not recognize that I've set the output as a function. Can someone spot my mistake? Code: Eqn1 = D[f[t], {t, 2}] + \[Omega]^2f[t] == 0; Sol1 = DSolve[Eqn1, f[t], t]; IC1 = f[0] == \[Alpha]; IC2 = (D[f[t], {t, 1}] == \[Gamma]) /. t -> 0; Sol2[t_] = Simplify[ DSolve[{Eqn1, IC1, IC2}, f[t], t] ]; x[t_] = Replace[ f[t], Sol2[t] ]; Manipulate[ Plot[ { x[t] }, { t, 0, 10 } ( Plot )], Style["", Bold, 14, FontFamily -> "Times"], {{\[Alpha], -10, Style["\[Alpha]", 16, FontFamily -> "Times"]}, -10, 10, 0.01, ImageSize -> Medium, Appearance -> "Labeled"}, "", Style["", Bold, 14, FontFamily -> "Times"], {{\[Gamma], 0, Style["\[Gamma]", 16, FontFamily -> "Times"]}, -10, 10, 0.01, ImageSize -> Medium, Appearance -> "Labeled"}, "", Style["", Bold, 14, FontFamily -> "Times"], {{\[Omega], 1.0, Style["\[Omega] [rad. \!\(\SuperscriptBox[\(s\), \(-1\)]\)]", 16, FontFamily -> "Times"]}, 0, 10, 0.01, ImageSize -> Medium, Appearance -> "Labeled"}, ControlPlacement -> Left, SaveDefinitions -> False, TrackedSymbols :> {\[Alpha], \[Gamma], \[Omega]} (* Manipulate *)]

I'm creating a template for differential equation solving. I'm trying to write a script that takes the output from the solution of the differential equation and write it as a new function to be used in Manipulate [ Plot[...] ]. However, when I do this, Plot will not recognize that I've set the output as a function. Can someone spot my mistake?

Code:

Eqn1 = D[f[t], {t, 2}] + \[Omega]^2*f[t] == 0;

Sol1 = DSolve[Eqn1, f[t], t];

IC1 = f[0] == \[Alpha];

IC2 = (D[f[t], {t, 1}] == \[Gamma]) /. t -> 0;

Sol2[t_] = Simplify[  DSolve[{Eqn1, IC1, IC2}, f[t], t]  ];

x[t_] = Replace[  f[t], Sol2[t]  ];


Manipulate[

     Plot[
      {
       x[t]
       },
      {  t, 0, 10  }
      (* Plot *)],

  Style["", Bold, 14, FontFamily -> "Times"],
  {{\[Alpha], -10, Style["\[Alpha]", 16, FontFamily -> "Times"]}, -10,
    10, 0.01, ImageSize -> Medium, Appearance -> "Labeled"},
  "",
  Style["", Bold, 14, FontFamily -> "Times"],
  {{\[Gamma], 0, Style["\[Gamma]", 16, FontFamily -> "Times"]}, -10, 
   10, 0.01, ImageSize -> Medium, Appearance -> "Labeled"},
  "",
  Style["", Bold, 14, FontFamily -> "Times"],
  {{\[Omega], 1.0, 
    Style["\[Omega] [rad. \!\(\*SuperscriptBox[\(s\), \(-1\)]\)]", 16,
      FontFamily -> "Times"]}, 0, 10, 0.01, ImageSize -> Medium, 
   Appearance -> "Labeled"},

  ControlPlacement -> Left,
  SaveDefinitions -> False,
  TrackedSymbols :> {\[Alpha], \[Gamma], \[Omega]}

  (* Manipulate *)]

POSTED BY: Tim Kirkpatrick

4 Replies

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Tim Kirkpatrick

Tim Kirkpatrick, Non Affiliated

Posted 11 years ago

POSTED BY: Tim Kirkpatrick

Henrik Schachner

Henrik Schachner, Radiation Therapy Center, Weilheim, Germany

Posted 11 years ago

Hi Tim, the reason is that variables controlled by Manipulate are local within Manipulate. To see this try: f[x_] := Sin[a x] Manipulate[a f[x], {x, 0, Pi}, {a, 1, 2}] Consequently the 'a' inside Sin and 'a' inside Manipulate are different variables. Cheers Henrik

POSTED BY: Henrik Schachner

Tim Kirkpatrick

Tim Kirkpatrick, Non Affiliated

Posted 11 years ago

That definitely fixed the problem. Thank you! I have absolutely no idea why that variable replacement in the Plot command corrects the issue though. Can you offer any feedback as to why this is the solution? Is the issue with Plot or Manipulate?

POSTED BY: Tim Kirkpatrick

Priyan Fernando

Posted 11 years ago

In the first part of the code you can replace the alphas, etc, by adding a 0 to the end: Eqn1 = D[f[t], {t, 2}] + \[Omega]0^2*f[t] == 0; Sol1 = DSolve[Eqn1, f[t], t]; IC1 = f[0] == \[Alpha]0; IC2 = (D[f[t], {t, 1}] == \[Gamma]0) /. t -> 0; Sol2[t_] = Simplify[DSolve[{Eqn1, IC1, IC2}, f[t], t]]; x[t_] = Replace[f[t], Sol2[t]]; Then this works: Manipulate[{Plot[ x[t] /. {\[Omega]0 -> \[Omega], \[Gamma]0 -> \[Gamma], \[Alpha]0 \ -> \[Alpha]} , {t, 0, 10}], x[1] /. {\[Omega]0 -> \[Omega], \[Gamma]0 -> \[Gamma], \[Alpha]0 -> \ \[Alpha]}}, {\[Omega], 1, 5, 1}, {\[Gamma], 1, 5, 1}, {\[Alpha], 1, 5, 1}] Should help you figure what was wrong.

In the first part of the code you can replace the alphas, etc, by adding a 0 to the end:

Eqn1 = D[f[t], {t, 2}] + \[Omega]0^2*f[t] == 0;

Sol1 = DSolve[Eqn1, f[t], t];

IC1 = f[0] == \[Alpha]0;

IC2 = (D[f[t], {t, 1}] == \[Gamma]0) /. t -> 0;

Sol2[t_] = Simplify[DSolve[{Eqn1, IC1, IC2}, f[t], t]];

x[t_] = Replace[f[t], Sol2[t]];

Then this works:

Manipulate[{Plot[
   x[t] /. {\[Omega]0 -> \[Omega], \[Gamma]0 -> \[Gamma], \[Alpha]0 \
->  \[Alpha]} , {t, 0, 10}], 
  x[1] /. {\[Omega]0 -> \[Omega], \[Gamma]0 -> \[Gamma], \[Alpha]0 -> \
 \[Alpha]}}, {\[Omega], 1, 5, 1}, {\[Gamma], 1, 5, 1}, {\[Alpha], 1, 
  5, 1}]

Should help you figure what was wrong.

POSTED BY: Priyan Fernando

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