ECH140ODESLaplaceTransform - Content-type...

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(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 135461, 4625] NotebookOptionsPosition[ 122859, 4300] NotebookOutlinePosition[ 123244, 4317] CellTagsIndexPosition[ 123201, 4314] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Solution of ODEs using Laplace Transforms", "Title", TextAlignment->Center, TextJustification->0, FontSize->24], Cell[CellGroupData[{ Cell["Copyright Brian G. Higgins (2010)", "Subsubsection", CellChangeTimes->{{3.474035702084041*^9, 3.474035702746293*^9}}], Cell[TextData[{ "All rights reserved. You may copy and modify these notebooks and its \ content only for internal use in your organization, provided that credit is \ given to Brian G. Higgins as the original author. All other uses require the \ written permission of the author. In particular, these notebooks and their \ content cannot be bought or sold or exchanged for profit, or incorporated \ into material that is bought or sold or exchanged for profit. The notebooks \ are provided \"as is\" without express or implied warranty. This notebook is \ compatible with ", StyleBox["Mathematica", FontSlant->"Italic"], " Version 7" }], "Text", CellChangeTimes->{{3.4740357182825117`*^9, 3.474035731778441*^9}}], Cell["Last Revision: January 2010", "Text", CellChangeTimes->{{3.474035705674842*^9, 3.4740357117222853`*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Subtitle"], Cell[TextData[{ "In this notebook we pull together all the properties of Laplace Transforms \ (see notebook on Laplace Transforms for details) to solve linear second order \ ODEs. We first discuss the basic strategy for finding a solution, and show \ how the calculations can be done analytically. Then we illustrate how ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used to solve the ODEs directly using the ", StyleBox["LaplaceTransform",
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FontWeight->"Bold"], " and ", StyleBox["InverseLaplaceTransform", FontWeight->"Bold"], " functions. We consider problems that have discontinuous forcing \ functions." }], "Text", CellChangeTimes->{3.4740379335283957`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[" Solution of ODEs", "Subtitle"], Cell["\<\ In this section we use all the concepts from the previous notebook on the \ properties of Laplace Transforms to solve ordinary differential equations. \ Consider the following generic second order inhomogeneous ODE with constant \ coefficients:\ \>", "Text", CellChangeTimes->{{3.474035762228441*^9, 3.474035771634595*^9}, 3.4740379427279043`*^9}], Cell[BoxData[ RowBox[{"\t\t\t", RowBox[{ RowBox[{ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["d", "2"], "y"}], SuperscriptBox["dt", "2"]], "+", RowBox[{ SubscriptBox["a", "1"], FractionBox["dy", "dt"]}], "+", RowBox[{ SubscriptBox["a", "0"], "y"}]}], "=", RowBox[{"g", RowBox[{"(", "t", ")"}]}]}], ",", " ", RowBox[{ RowBox[{"y", RowBox[{"(", "0", ")"}]}], "=", SubscriptBox["\[Beta]", "1"]}], ",", " ", RowBox[{ RowBox[{ FractionBox["dy", "dt"], RowBox[{"(", "0", ")"}]}], "=", SubscriptBox["\[Beta]", "2"]}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.474035762228441*^9, 3.474035769228685*^9}}], Cell["\<\ If we take the Laplace Transform (using the linearity property of the \ transform), the ODE becomes\ \>", "Text", CellChangeTimes->{{3.474035762228441*^9, 3.47403579396309*^9}}], Cell[BoxData[ RowBox[{"\t\t", RowBox[{ RowBox[{ RowBox[{"\[ScriptCapitalL]", RowBox[{"{", FractionBox[ RowBox[{
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SuperscriptBox["d", "2"], "y"}], SuperscriptBox["dt", "2"]], "}"}]}], "+", RowBox[{ SubscriptBox["a", "1"], "\[ScriptCapitalL]", RowBox[{"{", FractionBox["dy", "dt"], "}"}]}], "+", RowBox[{ SubscriptBox["a", "0"], "\[ScriptCapitalL]", RowBox[{"{", "y", "}"}]}]}], "=", RowBox[{"\[ScriptCapitalL]", RowBox[{"{", RowBox[{"g",
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  • Summer '13
  • LeThuy

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