DIGMath: Differential Equations – Florence S. Gordon

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Dynamic Investigations for Differential Equations using Excel

Sheldon P. Gordon and Florence S. Gordon, co-authors

Most of the following DIGMath (Dynamic Investigatory Graphics for Mathematics) programs require the use of macros to operate. In order to use these spreadsheets, Excel must be set to accept macros. To change the security setting on macros: When you open any of the spreadsheets, a new bar appears near the top of the window that says something like: “Security Warning: Active content has been disabled”, depending on the version of Excel you are using. Just click on Options, then click on “Enable the Content”, and finally click OK.

The following are the DIGMath investigations for Differential Equations that are currently (March, 2024) ready for use. (Several others are under development.) Please feel free to download and use any or all of these files. To access any individual files or the entire collection to download, click on this link: DIGMath for Differential Equations. The first item shown there is the associated zip file, DiffEqns.7z, containing all 18 current individual files. If you have problems downloading or running any of these files, please contact us at gordonsp@retiree.farmingdale.edu or flogo@optonline.net for assistance. If you have any suggestions for improvements or for new topics, please pass them on also.

1. The Slope, or Tangent, Field of a Differential Equation This DIGMath module allows you to investigate the slope field (also called the tangent field) associated with a differential equation of the form y‘ = f(x, y). You can enter your choice of function, and the window with x from xMin to xMax and y from yMin to yMax over which the tangent lines are to extend. The program then draws the associated slope field. Also, as you vary the coordinates of the initial point (x₀, y₀), it draws the graph of the corresponding solution, which you can see following the path determined by the tangent line segments.
2. Euler’s Method for Numerical Solutions to Differential Equations y’ = f(x, y) This DIGMath spreadsheet lets you investigate Euler’s Method for generating numerical approximations to the solution of the differential equation y‘ = f(x, y), for any desired function of x and y, with any desired initial condition x₀ and y₀. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, …, 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.
3. Euler’s Method for Numerical Solutions to Differential Equations y’ = f(x) This DIGMath spreadsheet lets you investigate Euler’s Method for generating numerical approximations to the solution of the differential equation y‘ = f(x), for any desired function of x (but not y) with any desired initial condition x₀ and y₀. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, …, 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.
4. Picard’s Method for Numerical Solutions to Differential Equation y’ = f(x, y) This DIGMath Module lets you investigate the successive approximations to the solution of the differential equation y’ = f(x, y) subject to the initial conditions x₀ and y₀ generated by Picard’s iterative method. You can select, by checking the appropriate boxes, which of the successive approximations (n = 1, 2, …, 6) you want displayed to better let you compare the convergence of the approximations. You can also select how far from x₀ you want the approximations to extend.
5. First Order Homogeneous Linear Differential Equations This DIGMath module lets you investigate the behavior of the solutions of the first order linear homogeneous differential equation y’ – ay = 0 with your choice of the coefficient a, and the initial condition y₀ at time t = 0. The spreadsheet allows you to explore the solution, both graphically and in closed form.
6. First Order Nonhomogeneous Differential Equations This DIGMath module lets you investigate the behavior of the solutions of the first order linear non-homogeneous differential equation y’ – ay = f(t), for a variety of basic families of functions — exponential, sine, cosine, linear, and quadratic. Using sliders, you can select the coefficient a in the differential equation and the parameters in the function you choose, as well as the initial condition y₀ at time t = 0. The spreadsheet displays the closed form expression for the solution, as well as the graph of the solution.
7. The Logistic Model This DIGMath module allows you to investigate visually two different aspects of the continuous logistic model based on the logistic differential equation P’ = aP – bP². (1) You can enter, via sliders, values for the two parameters a and b, as well as the initial population value P₀ and watch dynamically the effects on the resulting graph of the population, and also see the effects of changing any of these values. (2) You can also investigate visually the effects on the population of changes in the initial growth rate a and the maximum sustainable population (the limit to growth) L, along with the initial population value P₀, using sliders, and watch the dynamic effects on the graph of changing any of them.
8. The Predator-Prey Model This DIGMath spreadsheet lets you explore the Predator-Prey model that describes the interaction between two populations, one the predator (say, wolves) and the other the prey (say, rabbits). The model involves two interrelated differential equations, one for R ‘ and the other for W ‘. You can vary each of the four coefficients in the differential equations, as well as the initial values for the two populations to see the patterns over time. The spreadsheet also displays the phase plane portrait , which shows how one population changes with respect to the other population.
9. Newton’s Laws of Heating and Cooling This DIGMath module lets you explore both Newton’s Law of Heating and Newton’s Law of Cooling. Using sliders, you can enter the temperature of the medium, the heating or cooling constant (essentially, the rate at which the object heats up or cools), and the initial temperature of the object. The program draws the graph of the temperature function and allows you to trace along the curve to see the temperature value at different times.
10. Projectile Motion This DIGMath spreadsheet allows you to investigate the path of a projectile launched from ground level with initial velocity and initial angle of inclination α . You control the values for the initial velocity and the angle via sliders and the spreadsheet draws the path of the projectile, allows you to trace along the path via another slider, and displays the time t, the coordinates of the tracing point, and the vertical velocity at each point. One page uses the English system of measurements in feet and seconds and another page does the comparable displays in the metric system with centimeters and seconds. Among the suggested investigations is one involving finding the angle α for which the range of the projectile is maximum.
11. A Falling Object with Air Resistance This DIGMath module lets you investigate the situation where a falling object is subjected to air resistance, so the standard formulas for the height and velocity of the object as functions of time do not apply. Instead, you can investigate the two most widely used models with air resistance, one where the resistive force is proportional to the velocity v of the object and the other where the resistive force is proportional to the square of the velocity, v² . The spreadsheet lets you vary the initial height, the mass, and the resistive constant using sliders. It then displays the graph of the height as a function of time and the velocity of the object as a function of time.
12. Boundary Value Problems This DIGMath spreadsheet lets you explore the notion of a boundary value problem for the second order differential equation y” + ky = 0, whose solution is a sine curve. You can select either one page where you have the choice of an interval from t = 0 to an integer value or a second page where the interval extends from t = 0 to t = 2π . In each case, you have the choice of the coefficient k and the boundary values. The program displays the graph of the solution and the formula for the closed-form particular solution.
13. Second Order Homogeneous Differential Equations with Constant Coefficients This DIGMath module lets you investigate the behavior of the solutions of the second order homogeneous differential equation ay” + by’ + cy = 0 for your choice of the coefficients a, b, and c and initial conditions x₀, y₀, and y’₀.
14. Modeling a Spring This DIGMath module lets you investigate the behavior of a bob attached to a vertical spring. There are two options — no damping where the motion depends only on the mass of the bob, the initial displacement, and the spring constants or damping where the motion also depends on the viscous resistance coefficient. You can experiment with the effects of the coefficients in the case of simple harmonic motion (no damping) or the special cases of underdamping and overdamping when the resistive force is included.
15. The RLC Circuit Model This DIGMath program lets you investigate an RLC electric circuit, the most fundamental type of electrical circuit. It consists of three components, a resistor, whose resistance R is measured in ohms, an inductor, whose inductance L is measured in henries, and a capacitor, whose capacitance C is measured in farads. You can select the values for the inductance, the resistance, and the capacitance of the circuit, as well as the initial current. The spreadsheet then displays the graph of the current over time, which is the solution of the second order, linear homogeneous differential equation with constant coefficients L I” + R I‘ + (1/C) I = 0.
16. The Pendulum Model This DIGMath module provides a mathematical model for a pendulum – the motion of a bob hanging at the end of a string that oscillates back and forth repeatedly under the assumption that there is no air resistance that slows, or retards, the motion. You can select both the initial displacement angle θ₀ and the length L of the string, as well as the length of time that you want the process continued. The spreadsheet displays an animation of the motion and a display of the height of the bob as a function of time.
17. Orthogonal Trajectories for the Differential Equation y’ = f(x) This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ‘ = f (x). The orthogonal trajectories are the curves of solutions of the differential equation y ‘ = -1/f (x), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You can enter your choice of the function and the initial condition x₀ and y₀ , the extent over which solutions are displayed, and the choice of height for an intermediate solution curve.
18. Orthogonal Trajectories for the Differential Equation y’ = f(x, y) This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ‘ = f (x, y). The orthogonal trajectories are the curves of solutions of the differential equation y ‘ = -1/ f (x, y), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You can enter your choice of the function, the initial condition This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ‘ = f (x, y). The orthogonal trajectories are the curves of solutions of the differential equation y ‘ = -1/ f (x, y), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You enter your choice of the function, the initial condition x₀ and y₀ , the extent over which solutions are displayed, and the choice of height for an intermediate solution curve.

All of these files were developed under the support from a variety of grants from the National Science Foundation, to whom the authors are very appreciative.

To access any individual files or all of the files to download, click on this link: DIGMath for Differential Equations.