(************** Content-type: application/mathematica ************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 21352, 799]*) (*NotebookOutlinePosition[ 22212, 830]*) (* CellTagsIndexPosition[ 22168, 826]*) (*WindowFrame->Normal*) Notebook[{ Cell["Separable Differential Equations", "Subtitle", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["by George E. Hrabovsky", "Subsubtitle", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["The General Solution", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ We will frequently encounter differential equations which can be reduced to \ the following form,\ \>", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["A[x] dx + B[y] dy == 0\t\t\t\t(1)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ where A[x] is some function of x, and B[y] is some function of y. 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3*x - x^3) + (-2916 + (-54 + 216*(3*c - 3*x - x^3))^2)^(1/2))^(1/3)) + (-54 + 216*(3*c - 3*x - x^3) + (-2916 + (-54 + 216*(3*c - 3*x - x^3))^2)^(1/2))^(1/3)/ (6*2^(1/3))}, {y -> -1/2 - (3*(1 + I*3^(1/2)))/ (2*2^(2/3)*(-54 + 216*(3*c - 3*x - x^3) + (-2916 + (-54 + 216*(3*c - 3*x - x^3))^2)^(1/2))^(1/3)) - ((1 - I*3^(1/2))*(-54 + 216*(3*c - 3*x - x^3) + (-2916 + (-54 + 216*(3*c - 3*x - x^3))^2)^(1/2))^(1/3))/ (12*2^(1/3))}, {y -> -1/2 - (3*(1 - I*3^(1/2)))/ (2*2^(2/3)*(-54 + 216*(3*c - 3*x - x^3) + (-2916 + (-54 + 216*(3*c - 3*x - x^3))^2)^(1/2))^(1/3)) - ((1 + I*3^(1/2))*(-54 + 216*(3*c - 3*x - x^3) + (-2916 + (-54 + 216*(3*c - 3*x - x^3))^2)^(1/2))^(1/3))/ (12*2^(1/3))}} \ \>", "\<\ 1 {{y -> -(-) + 3 / 2 2/3 3 (2 Power[-54 + 216 (3 c - 3 x - x ) + 3 2 Sqrt[-2916 + (-54 + 216 (3 c - 3 x - x )) ], 1/3]) + 3 Power[-54 + 216 (3 c - 3 x - x ) + 3 2 1/3 Sqrt[-2916 + (-54 + 216 (3 c - 3 x - x )) ], 1/3] / (6 2 )}, 1 {y -> -(-) - (3 (1 + I Sqrt[3])) / 2 2/3 3 (2 2 Power[-54 + 216 (3 c - 3 x - x ) + 3 2 Sqrt[-2916 + (-54 + 216 (3 c - 3 x - x )) ], 1/3]) - 3 ((1 - I Sqrt[3]) Power[-54 + 216 (3 c - 3 x - x ) + 3 2 1/3 Sqrt[-2916 + (-54 + 216 (3 c - 3 x - x )) ], 1/3]) / (12 2 )}, 1 {y -> -(-) - (3 (1 - I Sqrt[3])) / 2 2/3 3 (2 2 Power[-54 + 216 (3 c - 3 x - x ) + 3 2 Sqrt[-2916 + (-54 + 216 (3 c - 3 x - x )) ], 1/3]) - 3 ((1 + I Sqrt[3]) Power[-54 + 216 (3 c - 3 x - x ) + 3 2 1/3 Sqrt[-2916 + (-54 + 216 (3 c - 3 x - x )) ], 1/3]) / (12 2 )}}\ \>"], "Output", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ This is an incredibly long and difficult solution involving imaginary \ numbers, a somewhat friendlier solution can be found by multiplying the \ results of the integration by 6,\ \>", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["6 x + 6 x^3/3 + 6 y^2/2 + 6 y^3/3 == 6 c\t(7)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Which yields,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["6 x + 2 x^3 + 3 y^2 + 2 y^3 == 6 c", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["which is the friendlier solution.", "SmallText", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Initial-Value Problem", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["There will be times when we have specific values of x and y,", \ "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["A[x] dx + B[y] dy == 0; y[x0] == y0\t\t(8)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["In this case we can integrate (8) in the following manner,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Integrate[A[t], {t, x0, x}] + \tIntegrate[B[s], {s, y0, y}] == 0\t(9)\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Example 1", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["Given the equation,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["Sin[x] dx + y dy == 0; y[0] == -2\t\t(10)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["Integrating (10) yields,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ Integrate[Sin[s], {s, 0, x}] + \tIntegrate[t, {t, -2, y}]\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ -1 + y^2/2 - 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6 y == 3 - 3 Exp[x^2].", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Equations Which Must be Separated", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Example 1", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["Given the equation,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["dy/dx == y/x^2\t\t\t\t\t(13)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ before we can integrate (13) we must first divide both sides by y and \ multiply both sides by dx,\ \>", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["dy/y == dx/x^2", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["and then subtract both sides by dx/x^2,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["dy/y - dx/x^2 == 0\t\t\t\t(14)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["now we can integrate both sides of (14),", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ Integrate[1/y, y] + \tIntegrate[1/x^2, x]\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ -x^(-1) + Log[y] \ \>", "\<\ 1 -(-) + Log[y] x\ \>"], "Output", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["Setting this equal to a constant c and then solving for y,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Solve[% == c, y]", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {{y -> E^((1 + c*x)/x)}} \ \>", "\<\ (1 + c x)/x {{y -> E }}\ \>"], "Output", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["Which is the solution of (13).", "SmallText", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Example 2", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["Given the equation,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["dy/dx == (x Exp[x])/(2 y)\t\t\t\t(15)", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["We must multiply both sides by (2 y) dx,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["2 y dy == x Exp[x] dx", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["We next subtract the right hand side from both sides,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["2 y dy - 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So the solution is,", "SmallText", ImageRegion->{{0, 1}, {0, 1}}], Cell["y + Log[y] == -7 + I Pi - x + 1/3 x^3.", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Conclusion", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Thus we end our discussion of separable differential equations, the simplest \ kind of ordinary differential equations. While there are many equations \ which can be solved in this way, the majority may not. 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