(*********************************************************************** 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). 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[ 8454, 223]*) (*NotebookOutlinePosition[ 9372, 253]*) (* CellTagsIndexPosition[ 9328, 249]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Math 325: Differential Equations, Assignment 1, February 11, \ 1998 Due to March 4, 1998.\ \>", "Subsection"], Cell["\<\ Names (up to 3 students): \ \>", "Subsection"], Cell[CellGroupData[{ Cell["Problem 1", "Subsubsection"], Cell[TextData[{ "In applying elasticity theory to study the transverse vibrations of a beam \ one encounters\nthe equation\n", StyleBox["\t(1) E*I*y''''[x] - l*b* y[x] == 0\n", "Input"], "where y[x] is related to the displacement of the beam at position x, the \ constant E is Young's modulus,\nI is the area moment of inertia which we \ assume is constant, l is the constant mass per unit length of\nthe beam, and \ b is a positive parameter to be determined. We can simplify the equation by \ letting \n ", StyleBox["(2)", "Input"], " ", StyleBox["K^4 = (l*b)/(E*I),\n", "Input"], "that is, we consider \n ", StyleBox["(3)", "Input"], " ", StyleBox["y''''[x] - K^4 y[x] == 0. \n", "Input"], "When the beam is clamped at each end, we seek a solution to equation (3) \ that satisfies the boundary conditions", StyleBox[" \n", "Input"], " ", StyleBox["(4)", "Input"], " ", StyleBox["y[0] = y'[0] = 0 and y[L] = y'[L] = 0 \n", "Input"], "where L is the length of the beam. The problem is to determine those \ values of K for which equation (3)\nhas a non trivial solution that satisfies \ (4). To do this proceed as follows:\n\n(a) Represent the general solution to \ (3) in terms of sines, cosines and exponentials.\n\n(b) Substitute the \ general solution obtained in part (a) into the equation in (4) to obtain four \ equations\nin four unknowns.\n\n(c) Determine those values of K for which the \ system of equations in part (b) has nontrivial solutions.\n" }], "Text"], Cell["\<\ Solution \ \>", "Subsubsection"], Cell["************************************************************", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2", "Subsubsection"], Cell[TextData[{ "Find the exact solution of the given initial value problem:\n\n ", StyleBox["(5)", "Input"], " ", StyleBox["y'[x] = 1/2 (y - 1)^2, y[0] = 1; x from [0,1]. ", "Input"], "\n\n Then use ", StyleBox["Mathematica", FontSlant->"Italic"], " program for the Euler's method twice to approximate this solution on the \ given interval, first with step size h = 0.01, then with step size h = 0.005. \ Then approximate the solution by using ", StyleBox["Mathematica", FontSlant->"Italic"], " function: NDSolve[]. Finally, make a table showing for each point x two \ approximate values and the actual value of y[x], together with the percentage \ error in the more accurate approximation, for x an integral multiple of 0.2." }], "Text"], Cell["\<\ Solution \ \>", "Subsubsection"], Cell["************************************************************", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 3", "Subsubsection"], Cell[TextData[{ "A skydiver jumps from an aircraft at an initial altitude of 10,000 ft. \ Assume that she falls vertically with\ninitial velocity zero v(0) = 0, \ weights m = 128 lb, and experiences an upward force F of air resistance given \ in terms of her velocity v(t) (in feet per second) by:\n\n", StyleBox["(6)", "Input"], " ", StyleBox["F = (0.01)v + (0.001)v^2 + (0.0001)v^3 ", "Input"], "\n\n(in pounds, and with the coordinate axis directed downward so that v > \ 0). \n\na) Find a differential equation for v = v(t) and formulate an initial \ value problem. \nHint: Use Newtons law: m a(t) = mg - F, where a(t) is an \ acceleration of the skydiver and\ng is acceleration of gravity.\n\ ******************************************************************************\ ******\nThe equation is as follows: v'[t] = f[v] where f[v] = 32 - F/4. Also \ v(0) = 0.\n\ ******************************************************************************\ ******\nb) The skydiver reaches her terminal velocity when the forces of \ gravity and air resistance balance, \nso f(v) = 0. (After that v[t] remains \ constant.) If she does not open her parachute, what will be her terminal \ velocity in this case? \nHint: Use ", StyleBox["Mathematica", FontSlant->"Italic"], " function: NSolve[a x^3 + b x^2 + c x + d == 0, x] for given coefficients\n\ a,b,c and d. Calculate Vterminal from f[v] = 0. Calculate .95*Vtermminal.\n\n\ c) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " program for the Runge-Kutta method to approximate solution to the initial\ \nvalue problem obtained in part (a). \n\n 1) How fast will a skydiver be \ falling after 5s have alapsed? After 10s? \n 2) How lond will it take to \ attain the terminal velocity?\n Hint: Use Runge-Kutta program and \ choose from the table of values t's for which v[t]'s\n are \ the closest to Vterminal and .95*Vtermminal.\n 3) How long will take to \ get to the 95% of the terminal velocity?\n 4) Sketch graph of the \ solution. ******** (By hand!) ************\n" }], "Text"], Cell["Solution", "Subsubsection"], Cell["************************************************************", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 4", "Subsubsection"], Cell[TextData[{ "a) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " program for the Runge-Kutta method to approximate solution to the \ following initial\nvalue problem :\n\n", StyleBox[ "(7) y'[x] = 5y[x] - 6 Exp[-x], y[0] = 1, x from [0,4]", "Input"], "\n\nwith step sizes h = 0.2, h = 0.1, and h = 0.05. Notice that the \ solution of (7) is: y[x] = Exp[-x]. \nMake a table showing for each point \ three approximate values and the actual value, for x an integral multiple of \ 0.4.\n\nIt is obvious from the table that our numerical approximation fails. \ The explanation lies in the fact that the general solution of the equation \ (7) is y[x] = Exp[-x] + C Exp[5x]. The particular solution satisfying the \ initial condition y[0] = 1 is obtained when C = 0. But any departure, however \ small, from the exact solution y[x] = Exp[-x] -- even if due only to roundoff \ error -- introduces (in effect) a nonzero value of C. And all solution curves \ with nonzero C diverge strongly away from the one with C = 0, even if their \ initial values are close to 1.\n\nb) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " function Plot [] to plot the solutions: y[x] = Exp[-x] + C Exp[5x], \n \ for C = 0, 0.05, 0.1, 0.2, 0.6 and demonstrate divergence of the solutions \ explained in part (a)." }], "Text"], Cell["\<\ Solution \ \>", "Subsubsection"], Cell["************************************************************", "Input"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, CellGrouping->Manual, WindowSize->{575, 655}, WindowMargins->{{104, Automatic}, {Automatic, 25}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002n88o?moogl" ] (*********************************************************************** 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. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 125, 4, 64, "Subsection"], Cell[1837, 55, 58, 5, 100, "Subsection"], Cell[CellGroupData[{ Cell[1920, 64, 34, 0, 42, "Subsubsection"], Cell[1957, 66, 1553, 30, 371, "Text"], Cell[3513, 98, 43, 4, 74, "Subsubsection"], Cell[3559, 104, 77, 0, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[3673, 109, 34, 0, 42, "Subsubsection"], Cell[3710, 111, 770, 16, 165, "Text"], Cell[4483, 129, 43, 4, 74, "Subsubsection"], Cell[4529, 135, 77, 0, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[4643, 140, 34, 0, 42, "Subsubsection"], Cell[4680, 142, 2100, 35, 499, "Text"], Cell[6783, 179, 33, 0, 42, "Subsubsection"], Cell[6819, 181, 77, 0, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[6933, 186, 34, 0, 42, "Subsubsection"], Cell[6970, 188, 1342, 24, 307, "Text"], Cell[8315, 214, 43, 4, 74, "Subsubsection"], Cell[8361, 220, 77, 0, 27, "Input"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)