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    Technique Narrative
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    5-005-StiffDifferentialEquations-TechniqueNarrative
    This material introduces the topic of ``stiffness'' for a system of ordinary differential equations (ODE's), through a series of examples. Stiffness is a property that a system of ODE's may posses that make it difficult to solve numerically.
    Euler's methodexplicit methodsimplicit methodsstiffsitffnessstiff differential equationinstability
    Modeling Scenario
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    4-020-AnIEDBlast-ModelingScenario
    These three exercises offer students a chance to model with second order ordinary differential equations, how they might incorporate a spring-mass system into a larger model, and how they can use the model to determine the results of a dynamical...
    applied mathematicsforcesFree Body DiagramImprovised Explosive Devicessum of forces
    Technique Narrative
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    1-030-RandomPerturbation-TechniqueNarrative
    After a brief historical view of this problem, we will demonstrate the derivation of first order linear differential equations with random perturbations.
    random perturbationBrownian motionLangevin equationRiemann-Steiltjes integralWiener processIto's calculus
    Modeling Scenario
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    1-081-TumorGrowth-ModelingScenario
    Students will transform, solve, and interpret a tumor growth scenario using non-linear differential equation models. Two population growth models (Gompertz and logistic) are applied to model tumor growth.
    logisticpopulationtumorGompertz
    Technique Narrative
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    7-005-OverViewLaplaceTransforms-TechniqueNarrative
    This is a specialized overview of Laplace Transform application to solving differential equations in Mathematica.
    integrationLaplace transformInverse Laplace transofrmintegration by parts
    Modeling Scenario
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    1-096-OP-AMP-Differentiator-ModelingScenario
    The output waveform (function) of a operational amplifier type of differentiator circuit is determined analytically from the first order governing ordinary differential equation and compared with the data acquired from numerical model (using...
    simulationMeasurementfeedbackMultisimcircuitvoltageoperational amplifiervalidationSPICE
    Technique Narrative
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    5-010-MatrixExponential-TechniqueNarrative
    The matrix exponential is a powerful computational and conceptual tool for analyzing systems of linear, constant coefficient, ordinary differential equations (ODE's). This narrative offers a quick introduction to the technique, with examples and...
    matrix esponentialPutzer's Algorithmdiagonalizationeigenvalueseigenvectors
    Modeling Scenario
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    1-104-InfectionRisk-ModelingScenario
    This project is designed to examine differences between the exponential and logistic growth models in biology and how to apply these models in solving epidemic questions.
    infectionlogisticexponentialcovidcarrying capacity
    Technique Narrative
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    9-001-SkinBurnModelNumericalMethods-TechniqueNarrative
    The heat equation is an important partial differential equation (PDE) which describes the distribution of heat in a given region over time. Numerical methods play an important role in solving these.
    Maternal-Fetal interfaceheat equationConservation of Energytheat fluxEuler's forward methodcentral differenceskin burnhyperthermiathermal conductivitylayers
    Modeling Scenario
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    1-137-SheepGraze-ModelingScenario
    In this activity, students will apply graphical analysis (such as phase lines) to determine the long-term predictions of a differential equation model for pasture grass using two different formulas for the herbivore consumption rate.
    herbivoreplantphase linesbifurcationoverexploitationsheepgraze
    Modeling Scenario
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    10-100-InsectOutbreaks-ModelingScenario
    We use a system of difference equations that incorporates a temperature-dependent MPB population growth rate to model the outbreak and recovery cycle in mountain pine beetle-infested forests.
    ecologyclimate changeinsectinsect outbreaksinsect outbreaksolitary wave
    Technique Narrative
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    5-030-LinNonHomoSystemSol-TechniqueNarrative
    We offer strategies for solving linear systems of nonhomogeneous differential equations using a conjectured solution strategy for a system of constant coefficient, linear, n
    strategiesdiscovery learningeigenvectorseigenvalue
    Modeling Scenario
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    6-070-BeerBubbles-ModelingScenario
    The goal of this project is to set up and numerically solve a first-order nonlinear ordinary differential equation (ODE) system of three equations in three unknowns that models beer bubbles that form at the bottom of a glass and rise to the top.
    molecular forensicsBEERspherebubbleNewton' Second LawIdeal Gas Lawdrag forceHadamardStokesmole
    Modeling Scenario
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    1-115-ModelingWithFirstOrderODEs-ModelingScenario
    Several models using first order differential equations are offered with some questions on formulating a differential equations model with solutions provided.
    bacteriafalling objectNewton's Law of Coolingdrugcoolingdrag
    Technique Narrative
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    1-009-Bifurcation-TechniqueNarrative
    We lead students to investigate first-order differential equations that contain unknown parameters. Students discover what happens to the qualitative behavior of solutions to these equations as these parameters vary.
    bifurcationqualitative behaviorbifurcation diagram
    Modeling Scenario
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    4-065-GasInjection-ModelingScenario
    Students use programs (or create their own code) based on exponential box-scheme approximations for solving systems of nonlinear differential equations that contain small parameters for the highest derivative terms or singularities in boundary...
    Prandtl-Blasius equationsboundary layergas injectionesponential box schemeuniform convergenceFORTRAN
    Modeling Scenario
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    3-045-RampBounce-ModelingScenario
    Students build two projectile motion models (1) a one-dimensional model for a vertically falling ball from a fixed distance until it hits an inclined ramp and (2) a two-dimensional projectile motion model of the ball bouncing off the ramp.
    projectile motioninclined rampcoefficient of restitutionoptimal anglebounce
    Article or Presentation
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    SIMIODE Spring 2024 Webinars - Insightmaker
    We discuss the use of the FREE system dynamics software Insightmaker (https://insightmaker.com/) in a first course in Ordinary Differential Equations (with a modeling emphasis).
    modelingsimulationSoftwarepythonsystemsInsightMakerflowCategory Theorystockstock-flow
    Technique Narrative
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    7-006-LaplaceTransformBirth-TechniqueNarrative
    We present a way of introducing the Laplace Transform as the continuous analogue of a power series expression of a function.
    Transformationspower series
    Technique Narrative
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    5-012-LinearSystemConjecture-TechniqueNarrative
    Students go from the solution for y'(t) = k*y(t) to a natural extension to the solution conjecture of a system of two constant coefficient, homogeneous, linear differential equations introducing eigenvalues and eigenvectors through student...
    discovery learningconjecturesolutionsubstitution