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Technique Narrative

142

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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

150

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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

441

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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

140

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108

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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

630

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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

165

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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

150

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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 method improved Euler's method RK3 methods RK4 methods order of accuracy absolute error CPU time

Technique Narrative

123

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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

110

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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

126

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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.

bifurcation qualitative behavior bifurcation diagram

Technique Narrative

136

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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

149

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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

105

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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 solution Euler's method improved Euler's method

Technique Narrative

118

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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

153

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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