# Math 216 Demonstrations

Book Sec Topic Model Type Example Description Link
1.1 1st order linear ODEs Simple ODE models A simple model for the velocity of a sprinter is presented, and solutions graphed. Sprinter
2.2 1st order linear ODEs Mixing problems A box with given volume and number of balls in it; every time interval some number of balls are added and some number removed. We model and simulate the number of balls. Mixing
2.3 Acceleration-velocity models Falling body with different friction models A model of a skydiver falling before and after parachute deployment. Skydiver
2.5 Population models, First-order autonomous ODEs First order population models A look at the logistic population model and a population model for the spruce budworm. For different initial conditions we look at the solution to the ODE, seeing the limiting value(s) of the population(s). Logistic
2.5 Population models, First-order autonomous ODEs First order population models A population model for the spruce budworm. For different parameter values, we look at the solution to the ODE, the graph of the RHS of the equation, and the phase plane. Budworm
Book Sec Topic Model Type Example Description Link
3.2 Systems of differential equations Two compartment model for drug intake A two compartment (GI tract and bloodstream) model for the amount of drug in a patient's system. The input function is either a step function or constant term, and transfers between compartments are taken to be proportional to the amount of the drug in the compartment. The solution to the system is graphed, or the solution along with the trajectory in the phase plane. Drugs
3.3 Solution of 2x2 systems (real eigenvalues) Simplified, two-compartment, model for lead in the body The three compartment model for lead in the body considered in Lead is simplified to give a two-compartment (and thus 2x2) system with "nice" eigenvalues and eigenvectors. The solution of the system with initial conditions is graphed and, if desired, shown as a trajectory in the phase plane. Lead2
3.4 Solution of 2x2 systems (complex eigenvalues) Interacting population model with harvesting A model for baleen whales (predator) and krill (prey), with harvesting, is linearized about the non-zero equilibrium solution to give a 2x2 system with complex eigenvalues. The solution of the system with given initial conditions is graphed and, if desired, shown as a trajectory in the phase plane. Whale
Book Sec Topic Model Type Example Description Link
4.1, 4.3 Solutions to linear, constant-coefficient, homogeneous differential equations Damped and undamped spring The angular position of a swinging door with a spring-loaded hinge is modeled, and the characteristic equation, roots of the characteristic equation, and solution to the differential equation shown simultaneously. SwingDoor
4.4 Mechanical vibrations Damped (and undamped) spring A simple mass-spring system with a dashpot is modeled. The behavior of the system and solution to the modeling equation are graphed simultaneously, and animated with the time as the animation parameter. Spring
4.5 Undetermined coefficients RLC circuit, radio tuner The RLC circuit is used as a model of a radio tuner. The response in the system is shown for a given circuit and input function, showing that (for the forcing functions for which the method of undetermined coefficients works) the response looks like the forcing function. Circuit
4.6 Response to sinusoidal forcing RLC circuit, radio tuner The RLC circuit model of a radio tuner is considered further, and the response to a given circuit and input function is considered. The response curves are considered as a function of the forcing frequency (or inductance), and the case of an input with multiple input frequencies used to show how one input frequency may be selected by the circuit. Response
Book Sec Topic Model Type Example Description Link
5.4 Laplace Transforms, step functions Acceleration/velocity model for a rocket lander A simple acceleration/velocity model for a rocket initially falling freely and subsequently firing a rocket to slow its deceleration to a speed more reasonable for a soft landing. The forcing is then a discontinuous function, which may be written with step functions and the resulting equation solved with Laplace transforms. Rocket
5.5 Laplace Transforms Two compartment model for drug intake A two compartment (GI tract and bloodstream) model for the amount of drug in a patient's system. The input function is a step function, possibly repeated, and transfers between compartments are taken to be proportional to the amount of the drug in the compartment. The system is rewritten as a single second-order equation that we might analyze with Laplace transforms and the solution graphed. Antihistamine
5.5 Laplace Transforms, step and periodic functions Two compartment model for drug intake This uses the model for the Antihistamine model, a two compartment model for the amount of drug in a patient's system, rewritten as a second-order equation. The forcing is a square wave, allowing the use of the transform of a periodic function to solve it. AntihistamineRepeated
Book Sec Topic Model Type Example Description Link
7.2 Linear analysis of nonlinear systems SI model for TB infection in Australian Opossums in New Zealand A model for tuberculosis infected Australian Opossums in New Zealand is considered in nondimensional form. There are two equilibrium solutions; phase portraits for the linearization around each are graphed, and those are put in the context of the phase portrait for the nonlinear system. Opossums
7.2 Linear analysis of nonlinear pendulum Mechanical systems model for a pendulum The equation modeling a simple nonlinear pendulum is considered, with an analysis of the linearization at different critical points being shown in the demonstration, along with the linear and nonlinear trajectories in the full phase plane. NLPendulum
7.3 Linear analysis of interacting species systems Interacting population model with harvesting A model for baleen whales (predator) and krill (prey), with harvesting, is linearized about equilibrium solutions. The solution of the linearized systems are graphed and the trajectories shown in the phase plane, along with trajectories from the nonlinear system. Whale2
Book Sec Topic Model Type Example Description Link
Lab 4 Numerical approximation: Euler's method Falling body, with numerical approximation A continuation of the skydiver model, with the addition of a numerical solution using Euler's method. Euler
Lab 4 Numerical approximation: Improved Euler and RK methods Falling body, with numerical approximation A further continuation of the skydiver model, with the addition of numerical solutions using the Improved Euler and Runge Kutta methods. NumMethod
Lab 4 Euler's method for systems Three compartment model for lead in the body A three compartment model for lead in the body is considered, with an approximate solution generated using Euler's method graphed over different timescales. Lead
Book Sec Topic Model Type Example Description Link
UMMath Math 216 Lecture Demos
Last Modified: Fri Sep 2 11:19:18 EDT 2016 (c)2013 Regents of the University of Michigan
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