(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@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[ 571573, 10195]*) (*NotebookOutlinePosition[ 572245, 10219]*) (* CellTagsIndexPosition[ 572201, 10215]*) (*WindowFrame->Normal*) Notebook[{ Cell["Forced Harmonic Oscillator", "Title"], Cell[BoxData[ FormBox[ FormBox[ RowBox[{ RowBox[{\(\(\[Omega]\^2\) \(y(x)\)\), "+", RowBox[{"2", " ", "b", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "(", "x", ")"}]}], "=", "0"}], "TraditionalForm"], TraditionalForm]], "NumberedEquation"], 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ImageRangeCache->{{{0, 431}, {265.875, 0}} -> {-1.07744, -1.12183, 0.0394228, 0.0126581}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[ "Lets add on a forcing term given by Sin[\[Omega]' t] on r.h.s of\n\ differential equation"], "Text"], Cell["The particular solution is", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(yp[t_, b_, \[Omega]_, \[Omega]d_] = \(\(-2\)\ b\ \[Omega]d\ Cos[\[Omega]d\ t] + \((\(-\[Omega]d\^2\) + \[Omega]\^2)\)\ Sin[\[Omega]d\ t]\)\/\(4\ b\^2\ \[Omega]d\^2 + \((\[Omega]d\^2 - \[Omega]\^2)\)\^2\)\)], "Input"], Cell[BoxData[ \(\(\(-2\)\ b\ \[Omega]d\ Cos[t\ \[Omega]d] + \((\[Omega]\^2 - \[Omega]d\^2)\)\ Sin[t\ \[Omega]d]\)\/\(4\ b\^2\ \[Omega]d\^2 + \((\(-\[Omega]\^2\) + \[Omega]d\^2)\)\^2\)\)], "Output"] }, Open ]], Cell["\<\ The general solution will be a sum of the characteristic solution and this particular solution. In order to satisfy the \ initial conditions we must find new c1 and c2 coefficients, however, for this example lets just use the c1 and c2 coefficients that we used for the unforced example, knowing that this implies a different set of initial conditions. 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