(*********************************************************************** 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[ 199208, 8580]*) (*NotebookOutlinePosition[ 200058, 8610]*) (* CellTagsIndexPosition[ 200014, 8606]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData["Math 2214, Gao"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Chapter 2, Algebraic Solution Methods"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Goal: To find exact solutions with ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->18, Background->RGBColor[1, 1, 0]], StyleBox["Mathematica", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->18, FontSlant->"Italic", Background->RGBColor[1, 1, 0]], StyleBox[".", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->18, Background->RGBColor[1, 1, 0]] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->18, Background->RGBColor[1, 1, 0]], Cell[TextData[ "Preparation: Read through the printed copy of this lab."], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->18, Background->RGBColor[1, 1, 0]], Cell[CellGroupData[{ Cell[TextData["Examples"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->18, FontSlant->"Italic"], StyleBox[" solves simple differential equations exactly.", Evaluatable->False, AspectRatioFixed->True, FontSize->18] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[TextData["Example 1. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[ "Find all solutions of the differential equation\nxy' + 2y = sin x, for x > \ 0."], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[CellGroupData[{ Cell[OutputFormData["\<\ DSolve[x*y'[x] + 2*y[x] ==Sin[x], y[x],x]\ \>", "\<\ DSolve[x y'[x] + 2 y[x] == Sin[x], y[x], x]\ \>"], "Input", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[OutputFormData[ "\<\ {{y[x] -> C[1]/x^2 + (-(x*Cos[x]) + Sin[x])/x^2}}\ \>", "\<\ C[1] -(x Cos[x]) + Sin[x] {{y[x] -> ---- + --------------------}} 2 2 x x\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->18] }, Open ]], Cell[TextData[ "The symbol C[1] stands for an arbitrary constant. \nThe output is in the \ form of a rule(note the arrow); we can define a function equal to the \ solution. (Note the format for defining a function.)\n \ "], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[TextData["Clear[y]"], "Input", AspectRatioFixed->True], Cell[TextData["y[x_, c_] := (c -x*Cos[x] + Sin[x])/x^2"], "Input", AspectRatioFixed->True], Cell[TextData[ "We now consider some particular solutions by fixing c."], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[CellGroupData[{ Cell[TextData["sol1=y[x,0]\nsol2=y[x,1]\nsol3=y[x,-3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (-(x*Cos[x]) + Sin[x])/x^2\ \>", "\<\ -(x Cos[x]) + Sin[x] -------------------- 2 x\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ (1 - x*Cos[x] + Sin[x])/x^2\ \>", "\<\ 1 - x Cos[x] + Sin[x] --------------------- 2 x\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ (-3 - x*Cos[x] + Sin[x])/x^2\ \>", "\<\ -3 - x Cos[x] + Sin[x] ---------------------- 2 x\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "We get some graphs that show the nature of these solutions for x>0."], "Text",\ Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[CellGroupData[{ Cell[OutputFormData["\<\ p1=Plot[sol1, {x,0,25}]\ \>", "\<\ p1 = Plot[sol1, {x, 0, 25}]\ \>"], "Input", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[TextData[ " -2\nPower::infy: Infinite expression 0. \ encountered."], "Message", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Infinity::indet: Indeterminate expression 0. 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