# Differential Equations Modules

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## Down to the Electricity & Magnetism Modules

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Introduction to boundary value problems and eigenvalue problems for ODEs in the context of buckling,heat conduction/diffusion,vibration problems. Intro to Fourier method.
 Authors: Richard Weinacht, George Hsiao, B. Lenhoff, Andrew Zydney, Jack R. Vinson Keywords: differential equations, heat transfer, diffusion, strength of materials, buckling, heat conduction, vibrating beam, vibrating string, BVP, eigenvalue Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

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The most important unit operation in a chemical process is generally a chemical reactor. Chemical reactions are either exothermic (release energy) or endothermic (require energy input) and therefore require that energy either be removed or added to the reactor for a constant temperature to be maintained. Exothermic reactions are the most interesting systems to study because of potential safety problems (rapid increases in temperature, sometimes called "ignition" behavior) and the possibility of exotic behavior such as multiple steady-states (for the same value of the input variable there may be several possible values of the output variable). In this module we consider a perfectly mixed, continuously stirred tank reactor (CSTR).
 Authors: B. Wayne Bequette Keywords: Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

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This module on lake pollution is about using differential equations to model lake pollution. To model lake pollution can be complicated, because there are many different factors which can be taken into account. We will use a simplified model, and consider only a few basic factors.
 Authors: Robert Wheeler, Kenneth Hannsgen, John Little, John Hoggard Keywords: differential equations Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

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There are many instances in biology and in the environment where mass transport is an important phenomenon. Everything from cellular osmosis to glucose uptake to pollution in lakes and ponds can be related to mass transport. The Mass Transport module introduces the ideas of concentration and concentration difference and formalizes how these concepts are responsible for many common phenomena. The module contains a series of related demonstrations and exercises that build on one another, allowing students an incremental understanding of the subject through increasing levels of complexity.
 Authors: Jonathan Newell, Russell Manson, Donald Drew, Jason Hylan, Elisa Barney Smith Keywords: concentration, differential equations, diffusion, dilution, mass transport, semi-permeable membrane, solute, solvent Learning Level: college Platform: Project Links base technical requirements Applets: Single Chamber, Two Sources to Single Chamber, Washout Dilution, Chamber and Sink, Two Chamber, Two Chambers Plus Sink, N Chamber (N=10), N Chamber (N=10) Plus Sink [ Full Metadata ]

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Linear systems of ODEs are introduced via the chemical reactions leading to the chlorination of benzene. One of the goals is to maximize the output of certain chlorinated benzene products.
 Authors: Gilberto Schleiniger, Andrew Zydney, L. P. Cook, B. Lenhoff Keywords: differential equations, linear systems, ODE, eigenvalues, eigenvectors, order reactions, initial value problems, kinetics Learning Level: college Platform: Project Links base technical requirements Applets: [ Full Metadata ]

## Mechanical Oscillations Series of Modules

A series of closely related modules addressing topics in mechanical oscillations, simple harmonic motion, and dynamic system behavior.

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Observations of a forced oscillator, physical and mathematical modeling of its dynamics, and comparisons between predicted and actual behavior.
 Authors: William Siegmann, William Boyce, Kevin Craig, Jessica Cavanaugh, Karen Leary, Celal Tufekci, Jeongmin Lee, Christine Goering Keywords: differential equations, mechanical oscillations, spring, mass, damper, damping, forced, forcing Learning Level: college Platform: Project Links base technical requirements Applets: Sine + Cosine, Beats, AF Plotter, Damped Forced Spring Mass, Undamped Forced Spring Mass [ Full Metadata ]

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Observations of a linear pendulums, physical and mathematical modeling of its dynamics, and comparisons between predicted and actual behavior.
 Authors: William Siegmann, William Boyce, Kevin Craig, Christine Goering Keywords: differential equations Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

### Multiple Spring Mass System

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Observations of systems with two masses and three springs, physical and mathematical modeling of their dynamics, comparisons between predicted and actual behavior, and extensions to systems with more masses and springs.
 Authors: William Siegmann, William Boyce, Kevin Craig, Les Rubenfeld, Peter Olson Keywords: differential equations Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

### Non-Linear Pendulum

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Observations of a non-linear pendulums, physical and mathematical modeling of its dynamics, and comparisons between predicted and actual behavior.
 Authors: William Siegmann, William Boyce, Kevin Craig, Christine Goering Keywords: differential equations Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

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Observations of a mass-spring system, physical and mathematical modeling of its dynamics, and comparisons between predicted and actual behavior.
 Authors: William Siegmann, William Boyce, Kevin Craig, James Fairweather, Brian Griglak, Celal Tufekci, Karen Leary, Jessica Cavanaugh, Jeongmin Lee, Franklin Turner, Rahul Kulkarni, Laura Gross, Christine Goering, Sean Russell Keywords: differential equations, mechanical oscillations, spring, mass, damper, damping Learning Level: college Platform: Project Links base technical requirements Applets: Undamped Spring Mass, Damped Spring Mass, Sine + Cosine [ Full Metadata ]

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Observations of a spring pendulum system, physical and mathematical modeling of its dynamics, and comparisons between predicted and actual behavior.
 Authors: William Siegmann, William Boyce, Kevin Craig, Gregor Kovacic, Billie Rinaldi, Greg Pavliotis Keywords: differential equations Learning Level: college Platform: Project Links base technical requirements Applets: [ Full Metadata ]

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Interactive simulation of the vibrating string; comparison with experimental data. Interactive derivation of the solution of the wave equation for the vibrating string.
 Authors: Gregor Kovacic, Robert Degeneff, Steve Braun, Billie Rinaldi Keywords: differential equations Learning Level: college Platform: Project Links base technical requirements Applets: [ Full Metadata ]

## Electricity & Magnetism Series of Modules

A series of closely related modules addressing topics in electrostatics, currents and circuits, magnetism, electromagnetic induction, electrodynamics and optics.

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This module will help you understand how a circuit with an inductor and a capacitor undergoes an electromagnetic oscillation, and how the period and frequency of an oscillation depend on the inductance and capacitance.  You should also finish with an understanding of how the energy in a circuit is conserved if there is no resistance.
 Authors: Mike Malak, Joe Haus Keywords: physics Learning Level: College Platform: Project Links base technical requirements Applets: [ Full Metadata ]

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This module will help you understand what electrical current and resistance are, the relationship between voltage, current, and resistance. You will also be able to calculate these quantities for resistors in a circuit, the resistance of a network of resistors, and the resistance of a resistor given its size and shape and the resistivity of its material.
 Authors: Mike Malak, Joe Haus Keywords: physics Learning Level: College Platform: Project Links base technical requirements Applets: Compound Resistor , Variable Resistor [ Full Metadata ]