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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
    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
    Technique Narrative
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    2-005-LinearizeItAll-TechniqueNarrative
    Linear approximations are often used to simplify nonlinear ordinary differential equations (ODEs) for ease in analysis. The resulting linear approximation produces an ODE where closed form solutions may be obtained.
    computationerrorTorricelli's Lawdata fittinglinear approximation
    Technique Narrative
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    2-001-NumericalMethodsComparisons-TechniqueNarrative
    This material teaches the basics of numerical methods for first order differential equations by following graphical and numerical approaches. We discuss the order of accuracy of the methods and compare their CPU times.
    Euler's methodimproved Euler's methodRK3 methodsRK4 methodsorder of accuracyabsolute errorCPU time
    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
    Technique Narrative
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    1-010-AtmosphericCO2Bifurcation-TechniqueNarrative
    Students are introduced to the concept of a bifurcation in a first-order ordinary differential equation (ODE) through a modeling scenario involving atmospheric carbon dioxide whish is taken as a parameter and temperature is a function of time.
    carbon dioxideSurface Atmosphere Exchangebifurcationfold bifurcationsaddle node
    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
    Technique Narrative
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    8-002-TrigSumRepresentation-TechniqueNarrative
    Students discover how to represent functions as sums of trigonometric functions and the value of such representations in many fields. This is an introduction to the study of Fourier Series.
    estimationTrigonometryTrigonometric Functionssum of square errorsfinite sumFourier seriesFouriere sumspectrumapproximatiopn
    Technique Narrative
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    1-005-NavigatingNumericalMethods-TechniqueNarrative
    This technique narrative is a discovery-based approach to learning the basics of numerical methods for first order differential equations, by following the graphical and analytical perspectives of the forward Euler method and second order Taylor...
    Euler's methodTaylor's Methodsearch and rescueCoaast Guardlost at sealongitudelatitudelimit definition
    Technique Narrative
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    155

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    1-003-IntroNumericalMethods-TechniqueNarrative
    We develop elementary approaches to numerically solving first order differential equations with Euler's Method, Improved Euler's Method and develop these geometrically to compute numeric solutions and compare them to analytic solutions.
    analytic solutionEuler's methodimproved Euler's method
    Technique Narrative
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    1-002-IntegratingFactor-TechniqueNarrative
    We develop a strategy to solve first order differential equations by transforming one side of the equation to the derivative of a product of two functions, thereby making it easy to antidifferentiate that side.
    Integrating Factor
    Technique Narrative
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    1-001-SepartionOfVariables-TechniqueNarrative
    We discuss strategies to solve first order, ordinary differential equations with mathematical models when the variables may be separated.
    solutionsstrategymethodapplications
    Modeling Scenario
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    3-043-BallisticModeling-SpongeDart-ModelingScenario
    The goal of this project is for students to develop, analyze, and compare three different models for the flight of a sponge dart moving under the influences of gravity and air resistance.
    projectile motionair resistanceballisticsdimensional analysisNewtonian mechanicssponge dartNerf gunNerf dartBuckingham's Theorem
    Modeling Scenario
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    1-082-MirroMirror-ModelingScenario
    This project models the ``Foucault Knife Edge Test,'' an optical test commonly used by amateur astronomers who make their own mirrors for reflecting telescopes. The goal of the test is to estimate the shape of the surface of a mirror from optical...
    astronomymirror climate change mitigation ocean deacidificationopticstelescopelensFoucaultinterpolation
    Modeling Scenario
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    1-097-SwimmingPool-ModelingScenario
    This project involves the dynamics of chlorine concentration during regular swimming pool maintenance cycles. Students will have the opportunity to use both analytic and numerical methods.
    concentrationcchlorinenumerical errorswimming poolhorineerreor
    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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    1-012-Sublimation-Modeling Scenario
    We offer data on the sublimation of dry ice (carbon dioxide) collected in a classroom setting so that students can model the rate of change in the mass of a small solid carbon dioxide block with a differential equation model, solve the...
    sublimationdry ice
    Modeling Scenario
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    1-034-T-FishMixing-ModelingScenario
    This activity gives students a chance to build the underlying differential equation and/or difference equation for a mixing problem using tangible objects (fish) and a student-designed restocking and fishing plan in a lake.
    simulationfishFishingmixingrestocking
    Modeling Scenario
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    1-033-SouthernBarbeque-ModelingScenario
    We offer raw data collected from two thermometers used in the smoking process of Southern barbecue. One thermometer measures the temperature inside of the smoke chamber and the other measures the internal temperature of the meat.
    nonlinear regressionlogisticNewton's Law of Cooling-Warmingbarbecue
    Modeling Scenario
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    1-029-ConeToCubeFlow-ModelingScenario
    An inverted right circular cone with a hole at the bottom is suspended above an open-topped cube which also has a hole in the center of the bottom. The cone is filled with water and we model water flow from cone to cube and out the bottom of the...
    waterflowTorricelli's Lawconecube