(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 583113, 11052] NotebookOptionsPosition[ 579161, 10914] NotebookOutlinePosition[ 579519, 10930] CellTagsIndexPosition[ 579476, 10927] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Solving Differential Equations", "Title", CellChangeTimes->{{3.622782406862731*^9, 3.622782416189557*^9}}], Cell[TextData[{ "An ordinary differential equation (ODE) is an equation involving a function \ y(x) and its derivatives y\[CloseCurlyQuote](x), y\[CloseCurlyQuote]\ \[CloseCurlyQuote](x), ... , ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["y", RowBox[{"(", "n", ")"}]], "(", "x", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ". The order of the ODE is the highest order of the derivatives occurring. \ For example, the ODE \ y\[CloseCurlyQuote]\[CloseCurlyQuote]\[CloseCurlyQuote](x)+2", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]], FormatType->"TraditionalForm"], "y\[CloseCurlyQuote]\[CloseCurlyQuote](x)+y(x)y\[CloseCurlyQuote](x)=", Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]], FormatType->"TraditionalForm"], " has order 3. A function y=y(x) which verifies this differential equation \ is called a solution of the ODE, or an integral curve. " }], "Subsection", CellChangeTimes->{{3.6992678785883408`*^9, 3.6992679744995747`*^9}, { 3.699268039463396*^9, 3.699268380066328*^9}, 3.699268439531056*^9}, FontColor->GrayLevel[0]], Cell[CellGroupData[{ Cell[TextData[{ "Example 1: \n", StyleBox["Solve the ODE ", FontWeight->"Plain", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox["dy", "dx"], "=", FractionBox[ SuperscriptBox["x", "2"], RowBox[{"1", "-", SuperscriptBox["y", "2"]}]]}], FontSize->36], TraditionalForm]], FontSize->24, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[" and sketch its solutions.", FontWeight->"Plain", FontColor->GrayLevel[0]] }], "Section", CellChangeTimes->{{3.622782459961516*^9, 3.622782565476468*^9}, { 3.622783029714426*^9, 3.6227830351620483`*^9}, {3.699268411916362*^9, 3.699268418755351*^9}, 3.6992684792069283`*^9}, FontWeight->"Bold"], Cell[TextData[{ "This ODE is of a special type and can be solved using separation of \ variables. We first write:\n", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["y", "2"]}]}], TraditionalForm]], FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[")dy = ", FontWeight->"Bold", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["dx", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" \n", FontWeight->"Bold"], "and integrate each side with respect to y and x.", StyleBox["\n", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{"\[Integral]", RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["y", "2"]}]}]}], TraditionalForm]], FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[")dy = ", FontWeight->"Bold", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{"\[Integral]", SuperscriptBox["x", "2"]}], TraditionalForm]], FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["dx", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" \n", FontWeight->"Bold"], "\nThe general solution is found (by integration) to be \n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"y", "-", FractionBox[ SuperscriptBox["y", "3"], "3"], "+", "c"}], "=", FractionBox[ SuperscriptBox["x", "3"], "3"]}], TraditionalForm]], FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0]], ", where c is a constant parameter." }], "Subsection", CellChangeTimes->{{3.6227840991408663`*^9, 3.6227842017184477`*^9}, { 3.622784280656753*^9, 3.622784299706006*^9}, {3.622784392617425*^9, 3.622784393504128*^9}, {3.62281324950844*^9, 3.622813301518928*^9}, { 3.653858541782558*^9, 3.653858551635207*^9}, {3.69926851807345*^9, 3.699268526807501*^9}, {3.699268617973092*^9, 3.6992687289826403`*^9}}, TextJustification->1.], Cell[TextData[{ "The implicit curves\n ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"y", "-", FractionBox[ SuperscriptBox["y", "3"], "3"], "+", "c"}], "=", FractionBox[ 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"SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"scalar\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Order: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Method: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Hermite\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Periodic: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic], BaselinePosition -> Baseline], DynamicModuleValues :> {}], StyleBox["]", "NonInterpretableSummary"]}]}, "CopyTag", DisplayFunction->(#& ), InterpretationFunction->("InterpolatingFunction[{{0., 5.}}, <>]"& )], False, Editable->False, SelectWithContents->True, Selectable->False]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.6538699694657373`*^9}] }, Open ]], Cell["\<\ The solution to the initial value problem from Example 2 is unique. In this \ case, the output of NDSolve is a nested list with one transformation rule. \ The transformation rule describes y as a (numerically computed) function of \ x. Conceptually, think of the transformation rule as something like y \ \[RightArrow] f[x_ ] \ \>", "Subsubsection", CellChangeTimes->{{3.653870104903838*^9, 3.6538701531640463`*^9}, { 3.6538702130997066`*^9, 3.6538703340096684`*^9}, {3.6538703920010157`*^9, 3.653870481700926*^9}, 3.653870582104106*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Recall : To apply a transformation rule (or a set of transformation rules) to \ an expression, we use the Replacement operator \[OpenCurlyDoubleQuote]/.\ \[CloseCurlyDoubleQuote] (slash-dot).\ \>", "Subsection", CellDingbat->"\[LightBulb]", CellChangeTimes->{{3.653870512627337*^9, 3.653870609703629*^9}, { 3.65387099260756*^9, 3.6538709941922617`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"y", "[", "2", "]"}], " ", "/.", " ", "sol1"}]], "Input", FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{"{", RowBox[{"-", "2.35757103190501`"}], "}"}]], "Output", CellChangeTimes->{3.653870641854548*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We can also plot the function returned by NDSolve[..], just as we plot any \ other function in the Wolfram language, using the command Plot[..]. Compare \ the graph below to the stream line that we had plotted in red in the last \ StreamPlot[..].\ \>", "Subsubsection", CellChangeTimes->{{3.653870666523717*^9, 3.6538708183493958`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "/.", "sol1"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "3"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}]], "Input", CellChangeTimes->{{3.622761297196371*^9, 3.62276138435019*^9}, { 3.622761419127385*^9, 3.622761465032783*^9}, {3.622761507652697*^9, 3.622761536943757*^9}, {3.622777467119959*^9, 3.62277748008809*^9}, 3.622788851635211*^9, {3.6228135047002707`*^9, 3.622813505772806*^9}, { 3.622813554892448*^9, 3.622813572324609*^9}, {3.62281361284494*^9, 3.622813631495583*^9}, {3.6228136754741917`*^9, 3.6228137843090687`*^9}, { 3.653868067369976*^9, 3.653868100372037*^9}, {3.6538699626540747`*^9, 3.653869965062252*^9}, {3.6538706602899513`*^9, 3.653870662049499*^9}}, 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ODE ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dx"], "=", FractionBox[ RowBox[{ RowBox[{"4", "x"}], "-", SuperscriptBox["x", "3"]}], RowBox[{"4", "+", SuperscriptBox["y", "3"]}]]}], TraditionalForm]], FontSize->36] }], "Section", CellChangeTimes->{{3.622790466683278*^9, 3.622790476625743*^9}, { 3.622790511711554*^9, 3.6227905335575237`*^9}, {3.6227906625672903`*^9, 3.622790669390012*^9}, {3.6538711094179993`*^9, 3.65387111076064*^9}, { 3.653874243652176*^9, 3.653874255935793*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Plot the vector field using ", StyleBox["StreamPlot[...]", FontWeight->"Bold"], " in the region x \[Element] [-3.5, 3.5], y \[Element] [-3.3, 3.3]. On the \ same plot, plot the trajectory starting at the point (0, 1) in Red, and the \ trajectory starting at the point (1, -1.5) in Green. " }], "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.622814564601383*^9}, 3.6228147114882507`*^9, {3.699270124745747*^9, 3.699270128856696*^9}}, TextJustification->1., FontSize->16], Cell[TextData[{ "Find the discontinuities of the vector field (where the direction of the \ arrows changes, and the slopes of the integral curves are \[Infinity] or -\ \[Infinity]) using ", StyleBox["NSolve[expr,vars]. ", FontWeight->"Bold", FontColor->GrayLevel[0]], "This command attempts to find numerical approximations to the solutions of \ the system expr of equations or inequalities for the variables vars." }], "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.622814588339142*^9}, { 3.6228146853418417`*^9, 3.6228146932370367`*^9}, {3.622816137363825*^9, 3.622816145483821*^9}, {3.699270157029698*^9, 3.69927016532058*^9}}, TextJustification->1., FontSize->16], Cell[TextData[{ "Use ", StyleBox["NDSolve[...]", FontWeight->"Bold"], " to solve the ODE from example 2, ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dx"], "=", FractionBox[ RowBox[{ RowBox[{"4", "x"}], "-", SuperscriptBox["x", "3"]}], RowBox[{"4", "+", SuperscriptBox["y", "3"]}]]}], TraditionalForm]], FontSize->24], ", with initial condition ", StyleBox["y(0)=1", FontWeight->"Bold"], ". The range for the variable x is assumed to be [-3,3]. Plot the function \ that NDSolve returns, on the domain [-3,3]." }], "ItemNumbered", CellChangeTimes->{{3.622814562760298*^9, 3.6228146100611067`*^9}, 3.6228147184322357`*^9, {3.699270280870914*^9, 3.699270280871196*^9}}, TextJustification->1., FontSize->16], Cell[TextData[{ "This ODE is a separable differential equation, that we can solve and get \ the family of solutions defined by the implicit equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[ RowBox[{"4", "y"}], FontSize->16], StyleBox["+", FontSize->16], RowBox[{ StyleBox[ FractionBox["1", "4"], FontSize->18], StyleBox[ SuperscriptBox["y", "4"], FontSize->16]}]}], StyleBox["=", FontSize->16], RowBox[{ StyleBox[ RowBox[{"2", SuperscriptBox["x", "2"]}], FontSize->16], StyleBox["-", FontSize->16], RowBox[{ FractionBox["1", "4"], StyleBox[ SuperscriptBox["x", "4"], FontSize->16]}], "+", "c"}]}], TraditionalForm]], FontSize->18, Background->RGBColor[0.88, 1, 0.88]], " . 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