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Math 224: Intro. Differential Equations, Modeling Approach
 
 
 

 

 
Exams

Midterm exams

Thr. 11:30-1pm

I. Feb. 23

II. Apr. 12

Final

May 4, 4-7pm

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Office hours:

Tue: 4-5pm

Wed: 3-4pm

Yost 331

Additional Help

 

Description

Differential equations (DE) and Calculus in general were develop in XVII-XVIII centuries by Newton, Euler, Lagrange (and their followers). They mark the advent of modern science. Since early days DEs were widely used in diverse fields of science, engineering, technology. Today the scope of applications include life and social sciences. In this course we shall use DEs, as a modeling tool for diverse applications. Along with the traditional analytic methods, we shall extensively use modern computer technology. Many software packages are available now for solving DE: analytically, graphically and numerically.

The traditional approach aims at analytic solutions (exact or approximate), studied by mathematical tools (Algebra, Calculus, Geometry). Such matematical methods (however advanced) have severe limitations. We shall many simple problems that arise in diverse applications, which allow no analytic solutions. Even when available, analytic solutions would often require additional work to make useful predictions, so one still needs computer tools.

In this course we shall combine analytic methods and computer tools to address many problems and systems, that lie beyond the traditional scope of DE/calculus classes. Mathematics (DEs) will be used as tool for modeling and exploration of real systems (physical, chemical, biological et al).

Students will learn

1. How to formulate DE- model for a particular system, process, phenomena

2. How to solve and analyze them by analytic methods and computater tools;

3. How to apply mathematics (DEs) to real data analysis and predictions, draw conclusions, address open-ended problems.

Many questions and problems in homework, exams, projects will ask to explain the meaning of your work, and draw conclusion. The projects will also require critical analysis and summary.


Computer tools

Many software packages have built-in codes for solving DE. They include:

1. Mathematica - a powerful, multipurpose software, that allows symbolic, numeric and graphic manipulation of data. A special add-on package VisualDSolve by S. Wagon is handy for visualization of DE solutions.

For a brief introduction to Mathematica. with demonstrations see Wolfram screencast.

2. MatLab is another multipurpose software.

Mathematica and Matlab are available to CWRU studnts at Software Center. You can use a software' of your choice, but computer will be essential for many homeworks / projects, and useful for other.

Text

Differential Equations, by Blanchard, Devaney and Hall, PWS.

Supplementary material (lecture notes, Mathematica notebooks, homework problems, solutions, tests and projects) are posted on web.

Notes include supplemental material, or different exposition from the book. Students are responsible for all this material.

No prior computer experience is expected, and most computer materials (Mathematica nb) is provided.

Lab projects

There will be few (1-2) special team-projects based on book labs and other. All projects require both mathematical (analytic) work and computation. The projects will be assigned in advance, least 3-4 weeks. For each project the team will write a report, that should include

Title Page, with authors, project title and date.

Abstract (no more than half-page long): a brief summary of the problem, method used to solve it, and your results.

Main body: (i) Description of the system, and mathematical formulation of your (DE) model, and problems; (ii) Analytic solution methods and results, when available; (iii) Numeric and graphic methods and results. Explain the significance, compare 'analytic' vs. 'numeric' approaches. Be concise and clear in your writing, label all formulae and annotate your figures or tables.

Conclusions: Explain the meaning and significance of the problem and results.

Appendix (optional) can include all technical details of calculation, derivation, and additional material needed for your work.

Syllabus

The syllabus contains a broad overview of topics, based on the textbook and additional notes.

Homework is assigned for each class, and collected for reveral topics on syllabus (to be announced). Selected problems are graded and solutions posted

Chapter review: for each chapter (1-6) students will write a 1-page review that should inclde:

  1. 2-3 important (interesting) models/examples from the chapter with proper details (physicxal system, DE model, variables, parameters)
  2. list a few basic concepts discussed in the chapter
  3. 2-3 solution methods/techniques used in the chapter illustrated with examples.

All (text, equations, plots) must fit within 1 page. legibly written or better typed.

The reviews are submitted with a midterm and include chapters covered in the test. They will be part of the test grade (20%)

Grade.

The grade in this course is based on two midterms (15% each), the final exam (30%), homework (30%)+ projects (10%).