(*********************************************************************** 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[ 7109, 250]*) (*NotebookOutlinePosition[ 7971, 278]*) (* CellTagsIndexPosition[ 7927, 274]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 325: Differential Equations\tAssignment 1\t\tFall 1997", "Subsection"], Cell["\<\ Names (up to 3 students): \ \>", "Subsection"], Cell["Section:", "Subsection"], Cell[CellGroupData[{ Cell["Problem 1", "Subsubsection"], Cell[TextData[{ "Verify that the functions\n", StyleBox[ "\ty1[t_] = 1; y2[t_] = t;\n\ty3[t_] = E^(-t); y4[t_] = t*E^(-t);\n", "Input"], "are solutions of the differential equation\n", StyleBox["\ty''''[t] + 2y'''[t] + y''[t] == 0\n", "Input"], "and calculate their Wronskian." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2", "Subsubsection"], Cell[TextData[{ "Show that the Wronskian of ", StyleBox["Cos[t]", "Input"], ", ", StyleBox["Cos[t]^3", "Input"], ", ", StyleBox["Cos[t]^5", "Input"], ", and ", StyleBox["Cos[5t]", "Input"], " is ", StyleBox["0", "Input"], ".\nEstablish this result without direct evaluation of the Wronskian." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 3", "Subsubsection"], Cell[TextData[{ "Find all the 10th roots of ", StyleBox["1024", "Input"], ", including complex numbers. Convert these to points in the plane and plot \ them." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 4", "Subsubsection"], Cell[TextData[{ "Find the roots of the characteristic equation and the corrseponding \ fundamental solutions to the differential equation\n", StyleBox[ "\ty''''[t] + 6y'''[t] + 17y''[t] + 22y'[t] + 14y[t] == 0\n", "Input"], "Superimpose plots of the fundamental solutions.Then find and plot the \ particular solution that satisfies the initial conditions:\n", StyleBox["\ty[0] == 1, y'[0] == 1, y''[0]==1, y'''[0] == 1\n", "Input"], "Compare your solution with the solution given by ", StyleBox["DSolve[]", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 5", "Subsubsection"], Cell[TextData[{ "Find the general solution to the following differential equation using the \ method of undetermined coefficents.\n", StyleBox[ "\ty''''[t] + 2y'''[t] + 5y''[t] + 8y'[t] + 4y[t]==Sin[2t] + t*E^(-t)\n", "Input"], "Superimpose plots of the particular solutions satisfying\n", StyleBox["\ty[0] == y0, y'[0] == 0, y''[0] == 0, y'''[0] == 0\n", "Input"], "for several values of ", StyleBox["y0", "Input"], ". Explain why all these solutions seem to pass through the ", StyleBox["t", "Input"], "-axis at the same points and at regular intervals for large values of ", StyleBox["t", "Input"], ". Try to determine the length of this common interval." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 6", "Subsubsection"], Cell[TextData[{ "Find approximate values of the solution of the initial value problem\n", StyleBox["\ty'[t] == (t^2 - y[t]^2) Sin[y[t]], y[0] = 1\n", "Input"], "using the Euler method with ", StyleBox["h = 0.1", "Input"], ", ", StyleBox["h = 0.05", "Input"], ", and ", StyleBox["h = 0.025", "Input"], ". Compare these results with the output of ", StyleBox["NDSolve[]", "Input"], " by superimposing plots of the data." }], "Text"], Cell["Solution", "Subsubsection"], Cell[TextData[{ "Hints: Define a function that does one step of the Euler method, e.g.,\n\t\ ", StyleBox["EMStep[{t_,y_}] = N[{t+h,y+h*f[t,y]}];", "Input"], "\nThen use ", StyleBox["NestList[EMStep, {t0,y0}, n]", "Input"], " to apply ", StyleBox["EMStep[]", "Input"], "\nsuccessively to its own output, starting at the point ", StyleBox["{t0,y0}", "Input"], " and continuing for ", StyleBox["n", "Input"], " steps.\nTo plot the results use ", StyleBox["ListPlot[...,PlotJoined->True]", "Input"], ". The output of ", StyleBox["NDSolve[]", "Input"], " is a list (in this case a list of one!) of substitution rules of the form\ \n", StyleBox["{y[t]->InterpolatingFunction[...][t]}", "Input"], ". To plot this, do something like\n", StyleBox["sol = NDSolve[...]; Plot[y[t]/.First[sol],{t,a,b}]", "Input"], ". You can combine plots with ", StyleBox["Show[]", "Input"], "." }], "Text"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 7", "Subsubsection"], Cell[TextData[{ "Find approximate values of the solution of the initial value problem\n", StyleBox["\ty'[t] == (t^2 - y[t]^2) Sin[y[t]], y[0] = 1\n", "Input"], "using the Runge-Kutta method with ", StyleBox["h = 0.1", "Input"], ", ", StyleBox["h = 0.05", "Input"], ", and ", StyleBox["h = 0.025", "Input"], ". Compare these results with the output of ", StyleBox["NDSolve[]", "Input"], " (from ", StyleBox["Problem 6", "Subsubsection"], ") by superimposing plots of the data." }], "Text"], Cell["Solution", "Subsubsection"], Cell[TextData[{ "Hints: Use the same ideas as in ", StyleBox["Problem 6", "Subsubsection"], ". For the Runge-Kutta step, use the formula on p.437 of the textbook. \ Several lines of calculation can be done at once by enclosing them in \ parentheses, e.g.,\n\t ", StyleBox["RKStep[{t_,y_}] = ( ...; ...; ...; N[{...,...}]);", "Input"] }], "Text"], Cell["", "Input"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, CellGrouping->Manual, WindowSize->{747, 640}, WindowMargins->{{20, Automatic}, {Automatic, 2}}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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