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    Technique Narrative
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    1-015-DimensionlessVariables-TechniqueNarrative
    This material introduces the idea of ``rescaling'' for ordinary differential equations (ODE's) by the use of dimensionless variables. In practice this is an extremely common and useful prelude to the analysis and solution of ODE's.
    linearization dimensional analysis time scale scaling dimensionless variables spatial scale
    Technique Narrative
    98

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    40

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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 interface heat equation Conservation of Energyt heat flux Euler's forward method central difference skin burn hyperthermia thermal conductivity layers
    Technique Narrative
    532

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    34

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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.
    Transformations power series
    Technique Narrative
    132

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    35

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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
    strategies discovery learning eigenvectors eigenvalue
    Technique Narrative
    119

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    30

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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 learning conjecture solution substitution
    Technique Narrative
    114

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    23

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    0

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    7-005-OverViewLaplaceTransforms-TechniqueNarrative
    This is a specialized overview of Laplace Transform application to solving differential equations in Mathematica.
    integration Laplace transform Inverse Laplace transofrm integration by parts
    Technique Narrative
    391

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    62

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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 esponential Putzer's Algorithm diagonalization eigenvalues eigenvectors
    Technique Narrative
    104

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    106

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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 method explicit methods implicit methods stiff sitffness stiff differential equation instability
    Technique Narrative
    600

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    40

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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 soluotion polynomial solutions Chebyshev polynomials Chebyshev differential equations
    Technique Narrative
    115

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    30

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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.
    computation error Torricelli's Law data fitting linear approximation
    Technique Narrative
    83

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    32

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    0

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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 perturbation Brownian motion Langevin equation Riemann-Steiltjes integral Wiener process Ito's calculus
    Technique Narrative
    77

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    65

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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 dioxide Surface Atmosphere Exchange bifurcation fold bifurcation saddle node
    Technique Narrative
    102

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    135

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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.
    estimation Trigonometry Trigonometric Functions sum of square errors finite sum Fourier series Fouriere sum spectrum approximatiopn
    Technique Narrative
    101

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    66

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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 method Taylor's Method search and rescue Coaast Guard lost at sea longitude latitude limit definition
    Technique Narrative
    101

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    31

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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.
    solutions strategy method applications