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    Modeling Scenario
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    3-030-SecondOrderIntro-ModelingScenario
    We outline the solution strategies involved in solving second-order, linear, constant coefficient ordinary differential equations, both homogeneous and nonhomogeneous and offer many application and modeling activities.
    parameter estimationnon-homogeneousSpanishfrequencybuoyancyinverse problemthrustunderdampedfirst passagespring mash dashpotrealcomplexhomogeneoussuperpositionparachuteoverdampedcritically dampedbuoyquadratic equationrootsrocket flight
    Modeling Scenario
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    3-040-FirstPassageTime-ModelingScenario
    We apply the notions of dampedness to second order, linear, constant coefficient, homogeneous differential equations used to model a spring mass dashpot system and introduce the notion of first passage time through 0 value with several applications.
    oscillatordampedunderdampedfirst passagefirst passage timespring mash dashpot
    Modeling Scenario
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    3-092-WirelessPower-ModelingScenario
    We present an engineering application (wireless power transmission) modelled by a coupled system of two linear second-order differential equations with constant coefficients. One equation is homogeneous while the other one is non-homogeneous.
    DESMOSwireless power transmissionelectric circuitsKirchhoff's Voltage Law
    Modeling Scenario
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    1-131-CaffeineElimination-ModelingScenario
    We model the concentration of caffeine eliminated from the human body at a rate proportional to the concentration. This is a ``first-order reaction'' in the language of pharmacokinetics -- the study of how drugs move in the body.
    caffeineeliminationpharmacokineticsdrug
    Modeling Scenario
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    1-015-Torricelli-ModelingScenario
    We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the column.
    dataSpanishtankcolumn of waterapertureTorricelli's LawConservation of EnergytSolverwater flowExcel Solver
    Modeling Scenario
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    5-001-LSD-ModelingScenario
    We describe the use of a two compartment model of a linear system of first order linear differential equations to model lysergic acid diethylamide (LSD) in the body. We provide the data from the literature.
    LSDSpanishcompartmentpharmacokineticslysergic acid diethylamide
    Modeling Scenario
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    1-070-FisheryHarvest-ModelingScenario
    Students model with logistic growth, harvesting, and diffusion in analyzing ocean fisheries of the Atlantic cod. We help students build models, ever more complex, to capture physical realities. At each stage we ask students to reflect on the model.
    logisticDiffusionmanagementharvestfishery
    Modeling Scenario
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    3-065-UpDown-ModelingScenario
    We model the height of a launched object which is subject to resistance proportional to velocity during its flight. We ask questions about the motion as well, e.g., highest point or apex and terminal velocity.
    terminal velocityNewton's Second Law of Motionvelocitymotionapex
    Technique Narrative
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    3-090-ChebyshevPolynomialSolution-TechniqueNarrative
    The Chebyshev equation is presented as a vehicle to view series solutions techniques for linear, second order homogeneous differential equations with non-constant coefficients.
    series soluotionpolynomial solutionsChebyshev polynomialsChebyshev differential equations
    Modeling Scenario
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    1-087-ThanosPopulationDynamics-ModelingScenario
    In the end of the “Avengers Infinity War,” the villain Thanos snaps his fingers and turns half of all living creatures to dust with the hope of restoring balance to the natural world. How does this affect the long term behavior of various species?
    logisticexponentialpopulationThanos
    Modeling Scenario
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    1-024-MalariaControl-ModelingScenario
    This project offers students a chance to make policy recommendations based on the analysis of models using both linear (exponential decay) and non-linear (logistic growth) differential equations.
    population dynamicsmalariaPolicyDecision-makingEuler's methodpharmacokineticsmosquitoimproved Euler's methodHuen's methodUS Army
    Modeling Scenario
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    1-017-DiseaseSpread-ModelingScenario
    Using a grid and m&m candies, we simulate the spread of disease. Students conduct the simulation and collect data to estimate parameters (in several ways) in a differential equation model for the spread of the disease.
    Modeling Scenario
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    1-014-DrainingContainers
    Given two rectangular circular cylinders of water with the same volume, but different radii, with a small bore hole of same radius on the center of the bottom through which water exits the cylinder, which empties faster?
    containerstankflowTorricelli's Lawdrainingfixed volume
    Modeling Scenario
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    1-005C-OilSlick-ModelingScenario
    We describe a modeling activity for Calculus I students in which modeling with difference and differential equations is appropriate.
    Problem Solvingoil slickspreadfittinglineassumptions
    Modeling Scenario
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    1-043-CoolingUpAndDown-ModelingScenario
    We consider modeling the attempt of an air conditioner to cool a room to a ``constant'' temperature.
    temperatureoscillationNewton's Law of Coolingcoolingthermostatair conditioner
    Modeling Scenario
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    1-045-TimeOfDeath-ModelingScenario
    Students are asked to determine the time of death given both environmental temperature situations and two observations of body temperature under several different circumstances.
    body masstemperaturedeathNewton's Law of Coolingbody temperature
    Modeling Scenario
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    1-063-ThreeHoleColumnOfWater-ModelingScenario
    We consider a column of water with three holes or spigots through which water can exit and ask students to model the height of the column of water over time.
    fallingcolumn of waterTorricelli's Lawmultiple spigotsspigots
    Modeling Scenario
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    1-064-TorricelliBox-ModelingScenario
    The time it takes a column of water to empty and the time it takes the same volume of column of water with a box (various sizes) submerged in the column of water are compared through modeling with Torricelli's Law.
    groundwater flowcolumn of waterTorricelli's Lawwater displacementsubmerged box
    Modeling Scenario
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    1-067-ModelingWithSigmoidCurves-ModelingScenario
    The assignment considers two well-known models of population growth, Verhulst-Pearl and Gompertz models, for which qualitative and quantitative analyses are provided. The graphs of the corresponding functions have a sigmoidal or S-shape.
    logisticGompertzsigmoidVerhulst-Pearl
    Modeling Scenario
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    1-068-WaterBottleCooling-ModelingScenario
    Students create of a differential equation describing how fluid in a water bottle will change its temperature to approach the ambient temperature in a room.
    bNewton's Law of Coolingwater bottleautonomousbottle