(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 5135016, 90691] NotebookOptionsPosition[ 5102608, 89717] NotebookOutlinePosition[ 5103720, 89754] CellTagsIndexPosition[ 5103496, 89746] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.3949044134974327`*^9, 3.394904422720999*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Physicist's Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " 7" }], "Title", CellChangeTimes->{{3.394579470096292*^9, 3.3945794805215673`*^9}, 3.395926458982993*^9, 3.441818231467331*^9}, FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["Fourth Edition 2007", FontSlant->"Italic"]], "Subtitle", CellChangeTimes->{{3.394579519605548*^9, 3.3945795336356897`*^9}, { 3.394579849753665*^9, 3.3945798497607594`*^9}, 3.4418182358126497`*^9}], Cell["PHYSICS 200", "Subtitle", CellChangeTimes->{{3.394579873314529*^9, 3.3945798826114264`*^9}}], Cell[CellGroupData[{ Cell["Nicholas Wheeler", "Author", CellChangeTimes->{{3.39457972270331*^9, 3.3945797314311037`*^9}}], Cell["REED COLLEGE", "Institution", CellChangeTimes->{{3.3945797341591387`*^9, 3.39457973632885*^9}, { 3.394579793724296*^9, 3.3945797937303047`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Author", CellChangeTimes->{{3.395183973403002*^9, 3.3951839734109497`*^9}}], Cell["", "Institution", CellChangeTimes->{{3.395183974221051*^9, 3.395183974224658*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Author", CellChangeTimes->{{3.3951839860326777`*^9, 3.3951839860365458`*^9}}], Cell["", "Institution"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Laboratory 4", FontColor->RGBColor[1, 0, 0]]], "Subtitle", CellChangeTimes->{{3.394581250479834*^9, 3.394581255896365*^9}, 3.394810647542058*^9, 3.394904595337902*^9, 3.3950042110191517`*^9, 3.39518399839618*^9}], Cell[CellGroupData[{ Cell["\<\ Obtaining/Displaying Solutions of Ordinary Differential Equations\ \>", "Section", CellChangeTimes->{{3.395184160768964*^9, 3.395184174408791*^9}, { 3.395190470255616*^9, 3.395190478912826*^9}, {3.395329163172098*^9, 3.395329175514009*^9}}], Cell[TextData[{ "Many problems in physics\[LongDash]as also elsewhere in the pure/applied \ mathematics of the last three centuries\[LongDash]assume this form: ", StyleBox["Find the ", FontColor->RGBColor[1, 0, 0]], StyleBox["x(t) ", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox["which, in collaboration with its derivatives, satisfies an \ equation of the form", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]] }], "Text", CellChangeTimes->{{3.395190498492792*^9, 3.3951905019118557`*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"F", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}], ",", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}], ",", "..."}], "]"}], "\[Equal]", "0"}], "Output", FontColor->GrayLevel[0.500008]]], "Input"], Cell[TextData[{ StyleBox["together with prescribed initial/boundary conditions", FontColor->RGBColor[1, 0, 0]], ". 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In the \ necessarily brief discussion that follows I will let physical problems \ provide my motivation." }], "Text", CellChangeTimes->{{3.395191307347562*^9, 3.39519132415539*^9}, { 3.395191403171393*^9, 3.395191421715074*^9}, {3.396050770911487*^9, 3.396050800745873*^9}}], Cell["", "Text", CellChangeTimes->{{3.39519154126648*^9, 3.3951915413743477`*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exponential Population Growth", FontColor->RGBColor[1, 0, 0]]], "Subsection", CellChangeTimes->{{3.395191592892686*^9, 3.395191603078629*^9}}], Cell[TextData[{ "Let ", StyleBox["P[t]", "Input"], " describe the size of a population at time ", StyleBox["t", "Input"], " . Population growth can, in the simplest model, be described" }], "Text", CellChangeTimes->{3.3960508427586937`*^9}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ RowBox[{"P", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{"a", " ", RowBox[{"P", "[", "t", "]"}]}]}], FontColor->GrayLevel[0.500008]]], "Output"], Cell[TextData[{ "The general solution is obvious, but let us ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to do the work:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"P", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{"a", " ", RowBox[{"P", "[", "t", "]"}]}]}], ",", RowBox[{"P", "[", "t", "]"}], ",", "t"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"P", "[", "t", "]"}], "\[Rule]", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"a", " ", "t"}]], " ", RowBox[{"C", "[", "1", "]"}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.4444833242099648`*^9}] }, Open ]], Cell[TextData[{ "The output is a ", StyleBox["replacement list", FontColor->RGBColor[0, 0, 1]], " (in this case, a one-element list, buried within a list). ", StyleBox["C[1]", "Output"], " is the name ", StyleBox["Mathematica", FontSlant->"Italic"], " gives to the arbitrary ", StyleBox["constant of integration", FontColor->RGBColor[0, 0, 1]], ". " }], "Text", CellChangeTimes->{{3.3951917956901093`*^9, 3.39519182056364*^9}}], Cell["", "Text", CellChangeTimes->{{3.395232286485011*^9, 3.395232286488823*^9}}], Cell[TextData[{ StyleBox["NOTE", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": The general solution of an ODE (or system of ODEs) is expected to contain \ constants of integration equal in number to the order of the ODE (or system). \ The present system is of first order, so we encounter only a solitary \ constant. " }], "Text", CellChangeTimes->{{3.395191858895392*^9, 3.395191989298658*^9}}], Cell["", "Text", CellChangeTimes->{{3.3952322952611027`*^9, 3.39523229526591*^9}}], Cell[TextData[{ "We could, at this point, reach inside the nested braces and ", StyleBox["by hand", FontVariations->{"Underline"->True}], " extract the information we seek, writing (say)" }], "Text", CellChangeTimes->{{3.395231136851555*^9, 3.3952311859820127`*^9}, { 3.395232325412355*^9, 3.3952323546414413`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"solution", "=", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"a", " ", "t"}]], " ", RowBox[{"C", "[", "1", "]"}]}]}]], "Input", CellChangeTimes->{{3.395231199296468*^9, 3.3952312140081997`*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"a", " ", "t"}]], " ", RowBox[{"C", "[", "1", "]"}]}]], "Output", CellChangeTimes->{3.4444833385634747`*^9}] }, Open ]], Cell["and verifying that indeed", "Text", CellChangeTimes->{{3.3952312229681387`*^9, 3.3952312279922028`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"D", "[", RowBox[{"solution", ",", " ", "t"}], "]"}], "\[Equal]", RowBox[{"a", StyleBox["*", "Code"], StyleBox["solution", "Code"]}]}]], "Input", CellChangeTimes->{{3.3952312435684547`*^9, 3.395231247346716*^9}, { 3.3952312880158367`*^9, 3.395231297988476*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.444483342421863*^9}] }, Open ]], Cell[TextData[{ "But, in more complicated situations, \"by-hand \ extraction\"\[LongDash]though recommended by its straightforward simplicity\ \[LongDash]can become a bit tedious. 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", StyleBox["ReplaceAll", "Input"], " (denoted ", StyleBox["/.", "Input"], ") we have already encountered, and works like this:" }], "Text", CellChangeTimes->{ 3.395232546391554*^9, {3.395232715269247*^9, 3.395232729026951*^9}, { 3.395232780986175*^9, 3.395232829493638*^9}, {3.3952328652460613`*^9, 3.3952328712812557`*^9}, {3.395232906273493*^9, 3.395232938131465*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"1", "+", SuperscriptBox["x", "2"]}], "/.", RowBox[{"x", "\[Rule]", "tom"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "b", ",", "x", ",", FractionBox["1", RowBox[{"1", "+", "x"}]], ",", "c"}], "}"}], "/.", RowBox[{"x", "\[Rule]", " ", "dick"}]}]}], "Input", CellChangeTimes->{{3.395232970877007*^9, 3.395232971440042*^9}}], Cell[BoxData[ RowBox[{"1", "+", SuperscriptBox["tom", "2"]}]], "Output", CellChangeTimes->{3.444483376534968*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"a", ",", "b", ",", "dick", ",", FractionBox["1", RowBox[{"1", "+", 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Simple \ pendula have here been demonstrated to be (because the torque is ", StyleBox["not a linear", FontSlant->"Italic"], " function of angle)", StyleBox[" anharmonic", FontColor->RGBColor[1, 0, 0]], "." }], "Text", CellChangeTimes->{{3.395359748522729*^9, 3.395359853139173*^9}, { 3.395359917184396*^9, 3.395359979801807*^9}}], Cell["", "Text", CellChangeTimes->{{3.395360030365078*^9, 3.395360030375348*^9}}], Cell["", "Text"], Cell[TextData[{ "\tIf you give a pendulum a strong enough initial push it will ", StyleBox["go right over the top", FontColor->RGBColor[0, 0, 1]], ": rather than \"oscillate\" it will loop-the-loop or \"librate\". 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