ORDINARY DIFFERENTIAL EQUATIONS This slide show consists of various examples from ODEs. When viewing the slides, the following keys are operational: HOME takes you to the first slide in the sequence you selected END takes you to the last slide in the sequence you selected UP ARROW takes you to the previous slide in the sequence you selected F9 immediately quits the program These keys do NOT operate like that while you are reading this document. A. One parameter family of curves 2 2 The function X + c/X is sketched for c = 1, .5, .25, 0, -.25, -.5, -1. B. The US Population and logistic growth The population of the US from 1790 is frequently modeled using logistic growth. Here these data are shown (in millions) for the period 1790 - 1950 in ten year intervals. Using 1790, 1850, and 1910 as exact values the appropriate logistic curve is drawn. Notice how well it fits the data points between 1790 and 1910, and how well it predicted the population until 1950 (with a minor perturbation in 1940 - was anything happening then to cause the population to drop?) Then the actual population from 1950 to 1980 is added to the first slide, and the same three data points are used, and the logistic curve is again overlaid. Surprise! For completeness here is the population of the US in millions from 1790 to 1980, in ten year intervals, taken from the World Almanac. 1790 3.929 1800 5.308 1810 7.240 1820 9.638 1830 12.861 1840 17.063 1850 23.192 1860 31.443 1870 38.558 1880 50.189 1890 62.980 1900 76.212 1910 92.228 1920 106.022 1930 123.203 1940 132.165 1950 151.326 1960 179.324 1970 203.302 1980 226.549 C. The cooling of coffee The temperature of a cup of coffee was recorded every minute for 14 minutes, when the ambient temperature was 22øC. The first slide shows this. Assuming Newton's law of cooling governs this process, the experimental results at 2 minutes, 14 minutes, and the ambient temperature were used to generate the theoretical curve in slide 2. Then a least squares fit was done on the experimental results at 2, 4, 6, 8, 10, 12, and 14 minutes, which were then used to produce the theoretical curve in slide 3. For completeness here is the coffee temp in degrees C, taken at one minute intervals (taken from Computer Simulation Methods, by Gould and Tobochnik, Addison-Wesley 1987). Time Temp 1 77.7 2 75.1 3 73.0 4 71.1 5 69.4 6 67.8 7 66.4 8 64.7 9 63.4 10 62.1 11 61.0 12 59.9 13 58.7 14 57.8 D. Numerical Methods - Euler This deals with solving the logistic equation y' = 10y(1-y) subject to the initial condition y(0) = .1 in the region 0 < x < 10. First the exact solution is displayed - notice that as x gets large the solution goes to 1. Then the numerical solution that would be obtained by using Euler's method is superimposed for different values of "h" (.18, .23, .25, .3). A bit worrying, isn't it! The idea for this demonstration comes from Fundamentals Of Differential Equations, by Nagle and Saff, Benjamin/Cummings 1989. Finally a slide is shown that explains what is behind this. It shows a plot of "h" versus the numerical solutions from h = .15 to h = .3. The idea for this demonstration comes from Chet Weiss, an undergraduate at the University of Arizona. E. Numerical Methods - Runge Kutta 4 This deals with solving the logistic equation y' = 10y(1-y) subject to the initial condition y(0) = .1 in the region 0 < x < 10. First the exact solution is displayed - notice that as x gets large the solution goes to 1. Then the numerical solution that would be obtained by using the Runga Kutta 4 method is superimposed for different values of "h" (.25, .3, .325. .35. .3675, .38). A bit worrying, isn't it! The idea for this demonstration comes from Fundamentals Of Differential Equations, by Nagle and Saff, Benjamin/Cummings 1989. Finally a slide is shown that explains what is behind this. It shows a plot of "h" versus the numerical solutions from h = .2 to h = .4. The idea for this demonstration comes from Chet Weiss, an undergraduate at the University of Arizona. F. Damped free vibrations The solutions of the differential equation X" + 2aX' + 64X = 0 subject to the initial conditions X(0) = 1, X'(0) = 0, are displayed for various values of the constant a. Critical damping occurs at a = 8. G. Series solution This is associated with the series solution of the differential equation xy" + y' + xy = 0 subject to the initial condition y(0) = 1. The idea is to build the function from the series. First we show the 1st term of the series expansion, then the 1st and 2nd terms, then the 2nd and 3rd terms, and so on. Can you see the function being created? In fact, you are looking at Jo(x), the Bessel function of the first kind of type 0, which is overlaid on the final slide. H. Bessel Function This graphs Jo(x), the Bessel function of the first kind of type 0, in the interval 0 < x < 8, and then overlays it with the polynomials 1 1 - x^2/4 1 - x^2/4 + (x^2/4)^2/(2!)^2 1 - x^2/4 + (x^2/4)^2/(2!)^2 - (x^2/4)^3/(3!)^2 1 - x^2/4 + (x^2/4)^2/(2!)^2 - (x^2/4)^3/(3!)^2 + (x^2/4)^4/(4!)^2 and so on, down to ... - (x^2/4)^7/(7!)^2 The radius of convergence of the Taylor series is infinity. When you have finished reading this document, press Q to quit.