(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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[ 110655, 3795]*) (*NotebookOutlinePosition[ 113347, 3863]*) (* CellTagsIndexPosition[ 112890, 3849]*) (*WindowFrame->Normal*) Notebook[{ Cell["First-Order Ordinary Differential Equations", "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has two routines that are designed to solve differential equations, ", StyleBox["DSolve", FontWeight->"Bold"], " and ", StyleBox["NDSolve", FontWeight->"Bold"], ". ", StyleBox["DSolve", FontWeight->"Bold"], " is used to find closed-form solutions, and ", StyleBox["NDSolve", FontWeight->"Bold"], " finds solutions numerically. Both can find solutions to ordinary as well \ as partial differential equations. See ", StyleBox["Mathematica", FontSlant->"Italic"], "'s Help Browser for syntax and other useful information." }], "Subsubsection", CellTags->{ "mmtag:20:ordinary_differential_equations__DSolve[]_and_NDSolve[]", "mmtag:20:numerical_solutions__ordinary_differential_equations"}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can solve the homogeneous linear ODE:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x]\ + \ p[x]\ y[x]\ \[Equal] 0, y[x], x]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \[ExponentialE]\^\(\[Integral]\_1\%x\(-p[K$26117]\) \ \[DifferentialD]K$26117\)\ C[1]}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The dummy integration variables and any integration constants are picked \ by ", StyleBox["Mathematica", FontSlant->"Italic"], " . Note that we asked ", StyleBox["Mathematica", FontSlant->"Italic"], " to solve the most general form of homogeneous linear first-order ODE, and \ we got a solution in a very general form that is equivalent to Kreyszig Eq. \ (3), p. 33. Here is a more specific problem and the solution found by ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x] + \((2\ x + 1)\) y[x]\ \[Equal] 0, \ y[x], x]\)], "Input"], Cell[BoxData[ \({{y[x] \[Rule] \[ExponentialE]\^\(\(-x\) - x\^2\)\ C[1]}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Boundary conditions are introduced in the following way to generate \ a particular solution:\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(y'\)[x] + \((2\ x + 1)\) y[x]\ \[Equal] 0, y[0] == 4}, \ y[x], x]\)], "Input", CellTags->"mmtag:20:DSolve[]__using_boundary_conditions"], Cell[BoxData[ \({{y[x] \[Rule] 4\ \[ExponentialE]\^\(\(-x\) - x\^2\)}}\)], "Output", CellTags->"mmtag:20:DSolve[]__using_boundary_conditions"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can solve the heterogeneous linear ODE:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x]\ + \ p[x]\ y[x]\ \[Equal] r[x], y[x], x]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \[ExponentialE]\^\(\[Integral]\_1\%x\(-p[K$26153]\) \ \[DifferentialD]K$26153\)\ C[ 1] + \[ExponentialE]\^\(\[Integral]\_1\%x\(-p[K$26153]\) \ \[DifferentialD]K$26153\)\ \(\[Integral]\_1\%x\( \[ExponentialE]\^\(-\(\ \[Integral]\_1\%K$26169\(-p[K$26153]\) \[DifferentialD]K$26153\)\)\ r[ K$26169]\) \[DifferentialD]K$26169\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Again, we asked ", StyleBox["Mathematica", FontSlant->"Italic"], " to solve the most general form of homogeneous linear first-order ODE, and \ we got a solution in a very general form that is equivalent to Kreyszig Eq. \ (4), p. 34. Here is a more specific problem and the solution found by ", StyleBox["Mathematica ", FontSlant->"Italic"], "(see Kreyszig Example 1, p. 34):" }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x]\ - \ y[x]\ \[Equal] \[ExponentialE]\^\(2 x\), y[x], x]\)], "Input"], Cell[BoxData[ \({{y[ x] \[Rule] \[ExponentialE]\^\(2\ x\) + \[ExponentialE]\^x\ C[ 1]}}\)], "Output"] }, Open ]], Cell[TextData[{ "The Bernoulli equation is a first-order ", StyleBox["nonlinear", FontSlant->"Italic"], " ODE that has a form that can be reduced to a linear ODE as described in \ Kreyszig, pp. 36\[Dash]38." }], "Text", CellTags->"mmtag:20:Bernoulli_equation"], Cell[CellGroupData[{ Cell[BoxData[ \(BernoulliEquation\ = \ \(y'\)[x]\ + \ p[x]\ y[x]\ == \ r[x]\ \((y[x])\)^\((a)\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(p[x]\ y[x]\), "+", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "\[Equal]", \(r[x]\ y[x]\^a\)}]], "Output"] }, Open ]], Cell[TextData[{ "The substitution ", "y(x) = ", Cell[BoxData[ \(TraditionalForm\`\((u \((x)\))\)\^\(1/\((1 - a)\)\)\)]], "is made, resulting in a linear ODE that can be solved for ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(yRep\ = \ u[x]\^\(1\/\(1 - a\)\)\), "\[IndentingNewLine]", \(DyRep\ = \ D[yRep, x]\)}], "Input"], Cell[BoxData[ \(u[x]\^\(1\/\(1 - a\)\)\)], "Output"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(DyRep\\)\\\" is similar to existing symbol \ \\\"\\!\\(yRep\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ FractionBox[ RowBox[{\(u[x]\^\(\(-1\) + 1\/\(1 - a\)\)\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], \(1 - a\)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(step1\ = BernoulliEquation /. {y[x] \[Rule] yRep, \(y'\)[x]\ \[Rule] \ DyRep}\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(p[x]\ u[x]\^\(1\/\(1 - a\)\)\), "+", FractionBox[ RowBox[{\(u[x]\^\(\(-1\) + 1\/\(1 - a\)\)\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], \(1 - a\)]}], "\[Equal]", \(r[x]\ \((u[x]\^\(1\/\(1 - a\)\))\)\^a\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(step2\ = PowerExpand[step1]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(p[x]\ u[x]\^\(1\/\(1 - a\)\)\), "+", FractionBox[ RowBox[{\(u[x]\^\(\(-1\) + 1\/\(1 - a\)\)\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], \(1 - a\)]}], "\[Equal]", \(r[x]\ u[x]\^\(a\/\(1 - a\)\)\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(step3\ = Simplify[step2]\)], "Input"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{\(u[x]\^\(-\(a\/\(\(-1\) + a\)\)\)\), " ", RowBox[{"(", RowBox[{\(\((\(-1\) + a)\)\ r[x]\), "-", \(\((\(-1\) + a)\)\ p[x]\ u[x]\), "+", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ")"}]}], \(\(-1\) + a\)], "\[Equal]", "0"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[step3, \(u'\)[x]]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\(-\((\(-1\) + a)\)\)\ \((r[x] - p[x]\ u[x])\)\)}], "}"}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "This last result is the first-order ", StyleBox["linear", FontSlant->"Italic"], " ODE that results from the Bernoulli equation. Its solution gives the \ function ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") which can be converted back to ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") with the relation ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = ", Cell[BoxData[ \(TraditionalForm\`\((u \((x)\))\)\^\(1/\((1 - a)\)\)\)]], "." }], "Subsubsection"], Cell["Numerical Solutions to ODEs", "Subtitle"], Cell["\<\ The following nonlinear ODE either does not have a solution, or \ takes too long to find.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[Sin[2\ Pi\ \(y'\)[x]^2]\ == \ y[x] x, y[x], x]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{"2", " ", "\[Pi]", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "2"]}], "]"}], "\[Equal]", \(x\ y[x]\)}], ",", \(y[x]\), ",", "x"}], "]"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ NDSolve is a numerical method for finding a solution. An initial \ condition and the desired range of solution are required.\ \>", \ "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(solution\ = NDSolve[{Sin[2\ Pi\ \(y'\)[x]^2]\ == \ y[x]\ x, y[0] == 1}, y, {x, 0, 3.5}]\)], "Input", CellTags->"mmtag:20:NDSolve[]"], Cell[BoxData[ RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]], \ "Message", CellTags->"mmtag:20:NDSolve[]"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 3.5`}}, "<>"]\), False, Editable->False]}], "}"}], ",", RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 3.5`}}, "<>"]\), False, Editable->False]}], "}"}]}], "}"}]], "Output", CellTags->"mmtag:20:NDSolve[]"] }, Open ]], Cell[TextData[{ "The results look kind of strange, perhaps, but they are a set of rules \ that provide a function that interpolates between values. \n Here is how \ to find the approximate solution at three different values of ", StyleBox["x", FontSlant->"Italic"], " on the specified interval:" }], "Text", CellTags->"mmtag:20:numerical_solutions_to_ODEs__interpolation_results"], Cell[CellGroupData[{ Cell[BoxData[ \(y[0.5]\ /. \ solution\)], "Input"], Cell[BoxData[ \({0.9074370101865736`, \(\(1.0973325823998172`\)\(\[InvisibleSpace]\)\) \ + 0.`\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y[1] /. solution\)], "Input"], Cell[BoxData[ \({0.744970458518884`, \(\(1.308772544883195`\)\(\[InvisibleSpace]\)\) - 0.014353625780704213`\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y[Pi] /. solution\)], "Input"], Cell[BoxData[ \({0.05249827705955056`, \(\(2.5018649128156945`\)\(\[InvisibleSpace]\)\) \ - 0.6106702474967095`\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[TextData[{ "Note that ", StyleBox["Mathematica", FontSlant->"Italic"], " has found two solutions, the first is real and the second is complex. \ Below are plots of the real and imaginary parts for both solutions:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[Evaluate[Re[y[x] /. solution]], {x, 0, 3.5}, PlotStyle \[Rule] {{Hue[1], Thickness[0.01]}, {Hue[0.6], Thickness[0.01]}}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.272109 0.0147151 0.216724 [ [.15986 .00222 -14.5 -20 ] [.15986 .00222 14.5 0 ] [.29592 .00222 -7 -20 ] [.29592 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