Re: Problem with InverseFunction
- To: mathgroup at smc.vnet.net
- Subject: [mg28208] Re: [mg28199] Problem with InverseFunction
- From: Mianlai Zhou <lailai at carmen.nikhef.nl>
- Date: Thu, 5 Apr 2001 03:00:24 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Hello, I think the reason of this is that Mathematica cannot solve this differential equation analytically, it can only express the result as the inverse function of something. So probably you will not be able to get rid of the "InverseFunction". And "#1" means the variable of the original function. Of course you can use the function Format to change the output style of the result. Good luck. On Wed, 4 Apr 2001, Low Choon Song wrote: > > Hello, > I have this problem that I have been trying to solve for weeks now. But I > still can't succeed. > I use DSolve in Mathematica to solve a Differential Eqn and in the output by > Mathematica, it always come out with the form of InverseFunction. > How can I express the results( output ) in a more readable form? > How can I get rid of the InverseFunction? > How can I get rid of the "#1" that appear in the solution? > > The problem I am solving is below and I have attached it too, if you are > willing to take a quick look. > Thanks in advance. > Low > > > (*********************************************************************** > > > Mathematica-Compatible Notebook > > This notebook can be used on any computer system with Mathematica 4.0, > MathReader 4.0, or any compatible application. The data for the > notebook > starts with the line containing stars above. > > To get the notebook into a Mathematica-compatible application, do one > of > the following: > > * Save the data starting with the line of stars above into a file > with a name ending in .nb, then open the file inside the application; > > * Copy the data starting with the line of stars above to the > clipboard, then use the Paste menu command inside the application. > > Data for notebooks contains only printable 7-bit ASCII and can be > sent directly in email or through ftp in text mode. Newlines can be > CR, LF or CRLF (Unix, Macintosh or MS-DOS style). > > NOTE: If you modify the data for this notebook not in a Mathematica- > compatible application, you must delete the line below containing the > word CacheID, otherwise Mathematica-compatible applications may try to > use invalid cache data. > > For more information on notebooks and Mathematica-compatible > applications, contact Wolfram Research: > web: http://www.wolfram.com > email: info at wolfram.com > phone: +1-217-398-0700 (U.S.) > > Notebook reader applications are available free of charge from > Wolfram Research. > ***********************************************************************) > > > (*CacheID: 232*) > > > (*NotebookFileLineBreakTest > NotebookFileLineBreakTest*) > (*NotebookOptionsPosition[ 9503, 185]*) > (*NotebookOutlinePosition[ 10219, 210]*) > (* CellTagsIndexPosition[ 10175, 206]*) > (*WindowFrame->Normal*) > > > > Notebook[{ > > Cell[CellGroupData[{ > Cell[BoxData[{ > \(\(Clear[Derivative, \ Rg, \ T, \ P, \ no, \ \ k, \ d, \ R, \ Ro, > \ n, \ > C\[Infinity], \ t];\)\), "\[IndentingNewLine]", > \(DSolve[{{\ \(R'\)[ > t]\ \[Equal] \ \(Rg\ T\)\/P\ \((\@\(2\ \((\(Rg\ T\ \ > C\[Infinity]\)\/P)\)\ \ d\ k\ 1\/R[t]\ + \ k\^2\/4\)\ \ - \ > 2\ k)\)\ \((\ \[Sqrt]\((1\/2\ \((P\/\(Rg\ T\))\) > \((\(k\ C\ > \[Infinity]\)\/\(\(\ \)\(d\)\))\)\ \((R[ > t])\)\ + \ \(\(\(1\)\(\ > \)\)\/\(\(16\)\(\ \ \)\ > \)\) \(\((P\/\(Rg\ T\))\)\^2\) \(\((k\/d)\)\^2\) \((R[t])\)\^2)\)\ - \ > \(\(1\ > \)\(\ \)\)\/\(\(4\)\(\ \)\)\ \((k\/d)\)\ \((P\/\(Rg\ T\))\)\ \((R[ > t]\ )\))\), \ R[0] \[Equal] \ 0}, \ R[t], \ t}, > \ > InverseFunctions\ \[Rule] \ False]\)}], "Input"], > > Cell[BoxData[ > \(DSolve::"argct" \(\(:\)\(\ \)\) > "\!\(DSolve\) called with \!\(2\) arguments."\)], "Message"], > > Cell[BoxData[ > RowBox[{"DSolve", "[", > RowBox[{ > RowBox[{"{", > RowBox[{ > RowBox[{"{", > RowBox[{ > RowBox[{ > RowBox[{ > SuperscriptBox["R", "\[Prime]", > MultilineFunction->None], "[", "t", "]"}], > "==", \(\(Rg\ T\ \((\(-2\)\ k + \@\(k\^2\/4 + > \(2\ C\ > \[Infinity]\ d\ k\ Rg\ T\)\/\(P\ R[t]\)\))\)\ \((\(-\(\(k\ P\ R[ > t]\)\/\(4\ d\ Rg\ T\)\)\) + > \@\(\(C\ > \[Infinity]\ k\ P\ R[t]\)\/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ > R[t]\^2\)\/\(16\ d\ > \^2\ Rg\^2\ T\^2\)\))\)\)\/P\)}], ",", \(R[0] == 0\)}], "}"}], ",", > \(R[t]\), > ",", "t"}], "}"}], ",", \(InverseFunctions \[Rule] > False\)}], > "]"}]], "Output"] > }, Open ]], > > Cell[CellGroupData[{ > > Cell[BoxData[ > RowBox[{"DSolve", "[", > RowBox[{ > RowBox[{"{", > RowBox[{ > RowBox[{ > RowBox[{ > SuperscriptBox["R", "\[Prime]", > MultilineFunction->None], "[", "t", "]"}], > "==", \(\(Rg\ T\ \((\(-2\)\ k + \@\(k\^2\/4 + \(2\ > C\[Infinity]\ > \ d\ k\ Rg\ T\)\/\(P\ R[t]\)\))\)\ \((\(-\(\(k\ P\ R[ > t]\)\/\(4\ d\ Rg\ T\)\)\) + \@\(\(C\ > \[Infinity]\ k\ P\ R[t]\)\/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ R[t]\^2\)\/\( > 16\ d\ > \^2\ Rg\^2\ T\^2\)\))\)\)\/P\)}], ",", \(R[0] == 0\)}], "}"}], ",", > \(R[t]\), > ",", "t"}], "]"}]], "Input"], > > Cell[BoxData[ > \({R[t] \[Rule] \(InverseFunction[ > C[1] - \((32\ d\ Log[ > 8\ C\[Infinity]\ d\ Rg\ T - > 15\ k\ P\ #1]\ \@\(\(8\ C\[Infinity]\ d\ k\ > Rg\ T + \ > k\^2\ P\ #1\)\/\(P\ #1\)\)\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ T\ #1 + > k\^2\ \ > P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\)\ \((\(-k\)\ P\ #1 + > d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ > T\ #1 \ > + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\))\)/\((225\ k\^3\ > \((8\ C\ > \[Infinity]\ d\ Rg\ T + > k\ P\ #1)\)\ \((\(-\(\(k\ P\ #1\)\/\(4\ d\ > Rg\ \ > T\)\)\) + \@\(\(C\[Infinity]\ k\ P\ #1\)\/\(2\ d\ Rg\ T\) + \(k\^2\ > P\^2\ \ > #1\^2\)\/\(16\ d\^2\ Rg\^2\ T\^2\)\))\))\) + \(1\/\(Rg\ T\)\) \((P\ \ > \((\(-\(\(8\ Log[\(-8\)\ C\[Infinity]\ d\ Rg\ T + > 15\ k\ P\ #1]\ \((\(-k\)\ P\ #1 + > > d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ > d\ k\ \ > P\ Rg\ T\ #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ > T\^2\)\))\)\)\/\(225\ k\^2\ > \ P\ \((\(-\(\(k\ P\ #1\)\/\(4\ d\ Rg\ T\)\)\) + \@\(\(C\[Infinity]\ k\ > P\ #1\ > \)\/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ > T\^2\)\))\)\)\ > \)\) - \(17\ Log[2\ \((4\ C\[Infinity]\ d\ Rg\ T + k\ P\ #1)\) + 2\ P\ > #1\ \@\ > \(\(8\ C\[Infinity]\ d\ k\ Rg\ T + k\^2\ P\ #1\)\/\(P\ #1\)\)]\ > \((\(-k\)\ P\ \ > #1 + d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ T\ #1 + k\^2\ P\^2\ > \ > #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\)\)\/\(225\ k\^2\ P\ \((\(-\(\(k\ P\ > #1\)\/\ > \(4\ d\ Rg\ T\)\)\) + \@\(\(C\[Infinity]\ k\ P\ #1\)\/\(2\ d\ Rg\ T\) + > \ > \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ T\^2\)\))\)\) + \((8\ \ > Log[\(-\(\(3375\ k\^2\ P\^2\ #1\ \@\(\(8\ C\[Infinity]\ d\ k\ Rg\ T + > k\^2\ P\ > \ #1\)\/\(P\ #1\)\)\)\/\(128\ C\[Infinity]\ d\ Rg\ T\ \((\(-8\)\ > C\[Infinity]\ > \ d\ Rg\ T + > 15\ k\ P\ #1)\)\)\)\) + \(3375\ > \((8\ \ > C\[Infinity]\ d\ k\^2\ P\ Rg\ T + 17\ k\^3\ P\^2\ #1)\)\)\/\(1024\ C\ > \[Infinity]\ d\ Rg\ T\ \((8\ C\[Infinity]\ d\ Rg\ T - 15\ k\ P\ > #1)\)\)]\ \((\ > \(-k\)\ P\ #1 + > d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ > k\ P\ \ > Rg\ T\ #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\))\)/\((225\ > k\^2\ \ > P\ \((\(-\(\(k\ P\ #1\)\/\(4\ d\ Rg\ T\)\)\) + \@\(\(C\[Infinity]\ k\ > P\ #1\)\ > \/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ > T\^2\)\))\))\) \ > - \((68\ Log[2\ \((4\ C\[Infinity]\ d\ Rg\ T + k\ P\ #1)\) + > 2\ d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ > d\ k\ \ > P\ Rg\ T\ #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\)]\ > \((\(-k\)\ P\ \ > #1 + d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ T\ #1 + k\^2\ P\^2\ > \ > #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\))\)/\((225\ k\^2\ P\ \((\(-\(\(k\ P\ > \ > #1\)\/\(4\ d\ Rg\ T\)\)\) + \@\(\(C\[Infinity]\ k\ P\ #1\)\/\(2\ d\ Rg\ > T\) + \ > \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ T\^2\)\))\))\) + \((32\ > Log[\(3375\ \ > \((8\ C\[Infinity]\ d\ k\^2\ P\ Rg\ T + 17\ k\^3\ P\^2\ > #1)\)\)\/\(4096\ C\ > \[Infinity]\ d\ Rg\ T\ \((8\ C\[Infinity]\ d\ Rg\ T - 15\ k\ P\ #1)\)\) > - \ > \(3375\ k\^2\ P\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ T\ #1 + k\^2\ > P\^2\ \ > #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\)\)\/\(512\ C\[Infinity]\ \((\(-8\)\ C\ > \[Infinity]\ d\ Rg\ T + 15\ k\ P\ #1)\)\)]\ \((\(-k\)\ P\ #1 + > d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ > k\ P\ \ > Rg\ T\ #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\))\)/\((225\ > k\^2\ \ > P\ \((\(-\(\(k\ P\ #1\)\/\(4\ d\ Rg\ T\)\)\) + \@\(\(C\[Infinity]\ k\ > P\ #1\)\ > \/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ > T\^2\)\))\))\) \ > - \(#1\ \((\(-k\)\ P\ #1 + d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ > Rg\ T\ \ > #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\)\)\/\(15\ > C\[Infinity]\ d\ > \ k\ Rg\ T\ \((\(-\(\(k\ P\ #1\)\/\(4\ d\ Rg\ T\)\)\) + > \@\(\(C\[Infinity]\ k\ > \ P\ #1\)\/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ > T\^2\)\ > \))\)\) - \(#1\ \@\(\(8\ C\[Infinity]\ d\ k\ Rg\ T + k\^2\ P\ > #1\)\/\(P\ #1\)\ > \)\ \((\(-k\)\ P\ #1 + d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ T\ > #1 + \ > k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\))\)\)\/\(60\ C\[Infinity]\ > d\ \ > k\^2\ Rg\ T\ \((\(-\(\(k\ P\ #1\)\/\(4\ d\ Rg\ T\)\)\) + > \@\(\(C\[Infinity]\ \ > k\ P\ #1\)\/\(2\ d\ Rg\ T\) + \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ > \ > T\^2\)\))\)\) - \(\@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ T\ #1 + k\^2\ > P\^2\ \ > #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\)\ \((\(-k\)\ P\ #1 + d\ Rg\ T\ \@\(\(8\ > C\ > \[Infinity]\ d\ k\ P\ Rg\ T\ #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ \ > T\^2\)\))\)\)\/\(15\ C\[Infinity]\ k\^2\ P\ \((\(-\(\(k\ P\ #1\)\/\(4\ > d\ Rg\ \ > T\)\)\) + \@\(\(C\[Infinity]\ k\ P\ #1\)\/\(2\ d\ Rg\ T\) + \(k\^2\ > P\^2\ \ > #1\^2\)\/\(16\ d\^2\ Rg\^2\ T\^2\)\))\)\) + \((\@\(\(8\ C\[Infinity]\ > d\ k\ \ > Rg\ T + k\^2\ P\ #1\)\/\(P\ #1\)\)\ \@\(\(8\ C\[Infinity]\ d\ k\ P\ Rg\ > T\ #1 \ > + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ T\^2\)\)\ \((\(-\(1\/\(60\ > C\[Infinity]\ > \ k\^3\ P\)\)\) + \(2\ d\ Rg\ T\)\/\(15\ k\^3\ P\ \((8\ C\[Infinity]\ > d\ Rg\ \ > T + k\ P\ #1)\)\))\)\ \((\(-k\)\ P\ #1 + > d\ Rg\ T\ \@\(\(8\ C\[Infinity]\ d\ > k\ P\ \ > Rg\ T\ #1 + k\^2\ P\^2\ #1\^2\)\/\(d\^2\ Rg\^2\ > T\^2\)\))\))\)/\((\(-\(\(k\ P\ > \ #1\)\/\(4\ d\ Rg\ T\)\)\) + \@\(\(C\[Infinity]\ k\ P\ #1\)\/\(2\ d\ > Rg\ T\) \ > + \(k\^2\ P\^2\ #1\^2\)\/\(16\ d\^2\ Rg\^2\ T\^2\)\))\))\))\) &]\)[ > t]}\)], "Output"] > }, Open ]] > }, > FrontEndVersion->"4.0 for Microsoft Windows", > ScreenRectangle->{{0, 800}, {0, 527}}, > WindowSize->{792, 500}, > WindowMargins->{{-3, Automatic}, {Automatic, -1}}, > PrintingCopies->1, > PrintingPageRange->{Automatic, Automatic} > ] > > > (*********************************************************************** > > Cached data follows. If you edit this Notebook file directly, not > using > Mathematica, you must remove the line containing CacheID at the top of > the file. The cache data will then be recreated when you save this > file > from within Mathematica. > ***********************************************************************) > > > (*CellTagsOutline > CellTagsIndex->{} > *) > > (*CellTagsIndex > CellTagsIndex->{} > *) > > (*NotebookFileOutline > Notebook[{ > > Cell[CellGroupData[{ > Cell[1739, 51, 781, 12, 158, "Input"], > Cell[2523, 65, 119, 2, 24, "Message"], > Cell[2645, 69, 785, 17, 99, "Output"] > }, Open ]], > > Cell[CellGroupData[{ > Cell[3467, 91, 650, 14, 96, "Input"], > Cell[4120, 107, 5367, 75, 1331, "Output"] > }, Open ]] > } > ] > *) > > > > > (*********************************************************************** > > End of Mathematica Notebook file. > ***********************************************************************) > >