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Go to The Start of ODE Lectures then scroll down to the current lecture video.

If you are using Mac go to Mac-Compatible Videos . You may have to download videos first and view them on Quicktime.

A short list of links of interest:

Direction Field Plotter John Polking, Rice University.

Slope Field Plotter. Solutions to many problems in Boyce, 7th ed, chapters 1,2,3,7,9,10 , David Schmidt at RPI

Course Information Spring 2013 Subject to change

A Guide to Basic Rules of Mathematical Writing

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Videos:

What you may be responsible for if you don't learn your ODE's!

Videos for solutions to many problems in Boyce, 7th ed, chapters 1,2,3,7,9,10 By David Schmidt at RPI

ODE at MIT. Video of lectures given by Arthur Mattuck and Haynes Miller, mathlets by Huber Hohn, at Massachussette Institute of Technology. Includes transcript, tools, exams, etc.

ODE Tools at JHU. Java tools for an ODE course given by Chikako Mese at John Hopkins University.

Interactive Differential Equations Addison Wesley Pearson set of applets for ODE.

Direction Field Plotter John Polking, Rice University.

Direction Field Software. Harmonic Oscillator Eric Woolgar, University of Alberta. Graph of solution of IVP ay"+by'+cy=0, y(0), y'(0).

Damped Pendulum J. Feldman, UBC.

Parametric Resonance J. Feldman, UBC.

A demo of Euler method J. Feldman, UBC.

Evolution with a random factor Paul Garrett, U Minnesota.

Differential Equations Solver for some famous equations with selections of RHS from a variety of inputs.

Symbolic Ordinary Equation Solver Robert Marik and Miroslava Tihlarikova .

Ordinary Differential Equation System Solver. Jens Langer, Jurgen Arndt, Felix Kramer. Technical University of Dresden.

IODE: Illinoise ODE UIUC matlab code for ODEs.

Differential Equations with Boundary Value Problems. software, homework, livemath by Selwyn Hollis. Use Internet Explorer and install LiveMath plugin. Or first make sure plugin works for your browser.

Nonlinear Pendulum Demo

Graphical solver for Second-Order ODEs. your variables are t, u, v=u'.

Texts:

Difference Equations to Differential EQuations Dan Sloughter.

Notes on Differential Equations Robert E. Terrell, Cornell.

Notes on Diffy Qs: Differential Equations for Engineers Jiri Lebel, U Wiskonsin Madison.

ODE notes By Paul Dawkins.

Applied Mathematics Saun Mauch. Section IV, Chapters 14-24.

A list of online texts in differential equations

Projects:

IDEA (Internet Differential Equation Activities: Student projects in ODEs)

Maple-based ODEs at U South Carolina Doug Meade.

Maple-based Honors ODEs at U South Carolina Doug Meade.

Mathematica based ODEs at U Manchester Gray, Mezzino,Pinsky.

ODE projects for Biology and Chemistry

Analyzers and Calculators:

Function Analyzer.

Wolfram Natural Language Portal for a Computational Knowledge Engine.

Wolfram Integrator.

A basic calculator.

Downloadable Software:

Scilab a free numerical package similar to Matlab.

Maxima a free symbolic computer algebra system .

Physics Applets:

physics applets Walter Fendt

physics applets Wolfgang Christian, Davison College.

physics applets Scott Schneider, LTU.

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Usage hints:

1- Consider using the fast forward button on Windows Media Player (WMP) to play the following videos with a speed up.

on WMP 10 you get speeds 1.4, 2, 5 by repeated clicks on fast forward button.

On WMP 11 one click on fastforward gets you speed 1.4 which is probably all you need, but you can click on "Now Playing" choose "Enhancements" Choose "Play Speed Settings" and adjust the speed up factor as you wish.

You may want to read about speed at Windows Media Player .

2- Videos before Section 2.4 are somewhat choppy.

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Review:

Study the hand-out and your calculus text to review basic rules of differentiation and integration.

Introduction, review of calculus 65MB. 90 min.

Go to calculus one and other topics for more detail.

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Study Section 1.1

Modeling of a fall. How a differential equation comes about. Falling object in presence of air friction. V'=g-(c/m)V, V= velocity, g=constant of gravity, c drag constant, m mass. 11MB.16 min.

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Direction Field, Equilibrium Solution. Graphical representation of a first order differential equation y'=f(t,y), by graphing a line segment with slope m=f(t,y) at point (t,y) in the coordinate plane of t and y. For now all examples are simple ones where t does not show up explicitly. 23MB.27 min.

Use Direction Field Plotter by John Polking, Rice University to help you draw direction fields. Wait for a minute for it to load the app, then click on DFIELD button, fill in the input box for x'=f(t,x).

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Long-Term behavior of solution Finding the asymptotic behavior of the solution using direction field. Example y'=y(y-3). 14MB. 23 min.

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Modeling of a mix. A mixture problem. A solution containing a dissolved substance enters a tank, gets mixed, and the mixed solution leaves the tank. Model and develop the differential equation governing the amount of dissolved subtance in the tank. 19MB. 25 min.

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Study Section 1.3

Classification of differential Equations to be uploaded

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Study Section 2.1

Integrating Factor Method 1. Solving first order linear equations y'+p(t)y=g(t). Define m=exp(int(p)), then y= [ int(mg)+C ]/m. Simple example given y'+3y=exp(t). Part of audio is lost. 29MB. 35 min.

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Study Section 2.2

Separable Equations y'=f(t,y), f is product or ratio a function of t and another function of y. 20MB. 25 min.

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Homogeneous Equations y'=f(t,y), where f is homogeneous of order zero, meaning it can be written in terms of y/t, or f(st, sy)= f(t,y) for all values of s. Can be converted to separable by using v=y/t. 10MB. 17 min.

Another example of homogeneous of order zero Solution of (x^2-2xy-y^2)y'=(x^2+2xy-y^2).

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Study Section 2.3

Modeling a Mix mixture problem with a variable volume, use of integrating factor method and initial condition. 34MB. 42 min.

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Study Section 2.4

Domain of Existence for Nonlinear Equations Influence of initial condition on the interval of existence. 26MB. 32 min.

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Domain of Existence for linear Equations. Uniqueness for linear and nonlinear equations. Two sample problems. 40 MB. 42 min.

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Study Section 2.5

Autonomous differential equations and population dynamics, Part 1 Qualitative description of solution of y'=(a-by)y, critical points, phase line, equilibrium solutions, stable, non-stable, semi-stable solutions, carrying capacity. 20 MB. 22 min.

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Autonomous differential equations and population dynamics, Part 2 Review of terminology: critical points, phase line, integral curves, equilibrium solutions, stable, non-stable, semi-stable solutions, determination through sign chart of f(y), horizontal translation of an integral curve produces another integral curve, integral curves do not collide. 20 MB. 22 min.

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Autonomous differential equations and population dynamics, Part 3 Parametric equations, analysis of graph of f(y) vs y, critical points where df/dy <0 are stable equilibrium solutions, df/dy>0 indicates instability, df/dy=0 requires further analysis, evan powers of factors indicate semi-stable and odd powers can be stable (crossing down) or unstable (crossing up). 20 MB. 22 min.

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Autonomous differential equations and population dynamics, Part 4 Schaefer Model for population with harvesting proportional to population, yield, optimal yield. 20 MB. 22 min.

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Autonomous differential equations and population dynamics, Part 5 Inflection point of integral curves y(t) is at max or min of graph of f(y) with respect to y which is typically where df/dy=0. 20 MB. 22 min.

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Study Section 2.6

Euler Method Approximate solution of first order equation y'=f(t,y) by a sequence of straight lines. 30 MB. 42 min.

Euler Applet by David Protas, CSU Northridge. Type in your function.

Euler Applet by Huber Hohn, MIT. Fixed function choice.

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Study Section 3.1

Second Order Differential Equations Introduction, general form, linear / nonlinear, constant coefficient / variable coefficient, homogeneous / non-homogeneous, initial conditions, IVP, linearity, superposition principle, 30 MB. 32 min.

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Study complex roots and repeated roots

Edition 8 Section 3.4, 3.5

Edition 9 Section 3.3, 3.4

2nd order, homogeneous, constant coefficient ODE Characteristic eqution, classification of roots, two distinct real roots, repeated roots, complex roots, euler formual, 30 MB. 32 min.

Study Section 3.4, 3.5

2nd order, homogeneous, constant coefficient ODE Characteristic eqution, classification of roots, two distinct real roots, repeated roots, complex roots, euler formual, 30 MB. 32 min.

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Study Reduction of order

Edition 8 Section 3.5

Edition 9 Section 3.4

Reduction of order method y = V * known solution, substitute in ODE to get a new ODE where V is missing, use W=V' to get a first order ODE, get W, integrate to get V, get y. 30 MB. 32 min.

Study Method of Undetermined Coefficient

another reduction of order example, one solution given, Problem 28 Page 173 Solve (x-1) y" - x y' + y = 0, y_1=e^x

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Study Method of Undetermined Coefficients

Edition 8 Section 3.6

Edition 9 Section 3.5

Method of Undetermined Coefficients, Part a Applicable to nonhomogeneous equations where source term is sum of products of polynomials, exponentials and sinusoidals. 30 MB. 32 min.

Method of Undetermined Coefficients, Part b More Examples. 40 MB. 32 min.

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Optional

Study Method of Variation of Parameters

Edition 8 Section 3.7

Edition 9 Section 3.6

Variation Of Parameters two solutions of corresponding nonhomogeneous equation are given we find the soltion of nonhomogeneous equation, Example Problem 5 Page 190 Solve y" + y = tan t , 0 < t < pi/2

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Study Mechanical Vibrations

Edition 8 Section 3.8

Edition 9 Section 3.7

Free Mechanical Vibrations vibrations of a spring, Newton's Law of motion, Hooke's Law, harmonic oscillation, derivation for frictioness or undamped oscillations. 30 MB. 32 min.

Free Mechanical Vibrations damped oscillattions, derivation. 30 MB. 32 min.

Visualization and Interactive Applets

Harmonic Oscillator Graph of solution of IVP ay"+by'+cy=0, y(0), y'(0). Eric Woolgar, University of Alberta.

Damped Mechanical Oscillations Oscillator, graph of roots of characteristic equations, graph of oscillation, slider for damping factor. Addison Wesley Pearson.

Damped Electrical Oscillations Addison Wesley Pearson.

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Study Mechanical Vibrations

Edition 8 Section 3.8

Edition 9 Section 3.7

Free Mechanical Vibrations Review of trigonometry, examples. 30 MB. 32 min.

Free Mechanical Vibrations Determination of mass, spring constant, and friction constant from given data. Internet-based resources.

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Study Mechanical Vibrations

Edition 8 Section 3.8

Edition 9 Section 3.7

Free Mechanical Vibrations Review of unit circle trigonometry, determination of phase shift. 30 MB. 32 min.

Free Mechanical Vibrations three examples.

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Study Forced Mechanical Vibrations

Edition 8 Section 3.9

Edition 9 Section 3.8

Forced Mechanical Vibrations Resonance.

Forced Mechanical Vibrations three examples.

Visualization and Interactive Applets

Movie of Tacoma Bridge destroyed by resonance.

amplitude resonse and resonance

Beats

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Study Forced Mechanical Vibrations

Edition 8 Section 3.9

Edition 9 Section 3.8

Forced Mechanical Vibrations examples, review.

Forced Mechanical Vibrations Beats.

Visualization and Interactive Applets

Beats

Sound of a beat Scott Schnider, LTU.

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Study Laplace Transform

Edition 8 Section 6.1

Edition 9 Section 6.1

Introduction to Laplace Transform basics, variety of transforms, kernel, improper integrals, piecewise defined functions, Laplace transform of f(t)=1.

Laplace transform of polynomials, exponentials and sinusiodals.

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Study Laplace Transform

Edition 8 Section 6.2

Edition 9 Section 6.2

Laplace Transform

Laplace Transform

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Study Laplace Transform

Edition 8 Section 6.2

Edition 9 Section 6.2

Laplace Transform

Corrections:

1- At time 15:20 the mutiple of ( t) is changed from (a) to (b). But on some formulas the change has not been carried out. In front of Sinh(bt) and Sin(bt) the numerator should read (b), and not (a).

2- At time 17:00 the right side of table needs the restriction s > a.

3- At time 32:50 instead of writing S+1 I write S-1. The error continues to the end of video. The problem is corrected and re-done in the next video.

Topic 1: Laplace[ exp(at) f(t)], several examples.

Topic 2: Partial Fraction Decomposition.

Topic 3: Completion of Squares.

Laplace Transform How to find inverse laplace transform in some basic cases. General 2nd order case.

Example 1: A linear term divided by a quadratic. Quadratic is irreducible, ie roots are complex.

Example 2: A quadratic divided by a cubic.

Example 3: Laplace transform of solution of ay"+by"+cy=g(t), y_0, y'_0.

Extra:

Laplace Transform Calculator. Xiao Gang, WIMS, France.

Laplace Transform in Wikipedia.

Inverse Laplace Transform Calculator. Rational functions only.

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Study Laplace Transform

Edition 8 Section 6.2

Edition 9 Section 6.2

Laplace Transform Laplace[t sin(at)], Laplace[(-t)^n f(t)]=F^(n) (s), Laplace transform of step/switch/Heaviside function u_c(t)

Study Laplace Transform

Edition 8 Section 6.3

Edition 9 Section 6.3

Laplace Transform external input with switch and delay, Laplace[u_3(t) (t-3)^2], writing multi-section oe piecewise-defined functions with add and delete approach.

Corrections:

At time 6:00 I write y=x^2. It should be y=t^2.

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Study Laplace Transform

Edition 8 Section 6.3

Edition 9 Section 6.3

Laplace Transform Laplace transform of a piecewise defined function, reformating functions by incorporating artificial delay

Corrections:

Cursor has not functioned properly on this recording.

At time 4:35 to 5:35. I write y(0), it should be f(0), also y'(0) should be f'(0), and y"(0) should be f"(0).

At time 14:48 I write an extra (t-2). It is corrected at 15:40 .

At time 22:30 I write F(t), it should be F(s).

Laplace Transform Two basic problems, 13 and 21 from 6.2.

Laplace Transform an example of finding Laplace[y(t)] for IVP with discontinuous forcing function, problem 24 from 6.2.

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Study Laplace Transform

Edition 8 Section 6.3

Edition 9 Section 6.3

Laplace Transform a review.

Laplace Transform a review.

Usage hints:

1- Consider using the fast forward button on Windows Media Player (WMP) to play the following videos with a speed up.

on WMP 10 you get speeds 1.4, 2, 5 by repeated clicks on fast forward button.

On WMP 11 one click on fastforward gets you speed 1.4 which is probably all you need, but you can click on "Now Playing" choose "Enhancements" Choose "Play Speed Settings" and adjust the speed up factor as you wish.

You may want to read about speed at Windows Media player .

Review:

Study the hand-out and your calculus text to review basic rules of differentiation and integration.

Introduction, review of calculus 65MB. 90 min.

And refresh your mind over a sample of differentiation and integration formulas.

Go to calculus one for more detail.

Start Studying 1.1

Introduction to Ordinary Differential Euations (ODEs) ODE as a relationship between a function and its derivative. Similarities between a direction field and a magnetic field.

Direction field, DField software, Connection to physics

Use Direction Field Plotter by John Polking, Rice University to help you draw direction fields. Wait for a minute for it to load the app, then click on DFIELD button, fill in the input box for x'=f(t,x). He uses x instead of our more traditional y.

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