Overview[ edit ] System dynamics is a methodology and mathematical modeling technique to frame, understand, and discuss complex issues and problems. Originally developed in the s to help corporate managers improve their understanding of industrial processes, SD is currently being used throughout the public and private sector for policy analysis and design. SD models solve the problem of simultaneity mutual causation by updating all variables in small time increments with positive and negative feedbacks and time delays structuring the interactions and control.
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Differential Equations Here are my notes for my differential equations course that I teach here at Lamar University.
Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.
In general, I try to work problems in class that are different from my notes. However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go.
With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Sometimes questions in class will lead down paths that are not covered here.
You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are. This is somewhat related to the previous three items, but is important enough to merit its own item. Using these notes as a substitute for class is liable to get you in trouble.
As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.
Here is a listing and brief description of the material that is in this set of notes. Basic Concepts - In this chapter we introduce many of the basic concepts and definitions that are encountered in a typical differential equations course.
We will also take a look at direction fields and how they can be used to determine some of the behavior of solutions to differential equations. Definitions — In this section some of the common definitions and concepts in a differential equations course are introduced including order, linear vs.
Direction Fields — In this section we discuss direction fields and how to sketch them. We also investigate how direction fields can be used to determine some information about the solution to a differential equation without actually having the solution. Final Thoughts — In this section we give a couple of final thoughts on what we will be looking at throughout this course.
First Order Differential Equations - In this chapter we will look at several of the standard solution methods for first order differential equations including linear, separable, exact and Bernoulli differential equations.
In addition we model some physical situations with first order differential equations. Linear Equations — In this section we solve linear first order differential equations, i. We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process.WRITING Describe three ways to solve a system of linear equations.
In Exercises 4 – 6, (a) write a system of linear equations to represent the situation. Then, answer the question using (b) a table, (c) a graph, and (d) algebra.
A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later. Please report any errors to me at [email protected] Writing Equation from Table of Values Often, students are asked to Write the equation of a line from a table of values.
To solve this kind of problem, simply chose any 2 points on the table and follow the normal steps for writing the equation of a line from 2 points. • Determine if an equation or inequality is appropriate for a given situation. • Solve mathematical and real-world problems with equations.
• Represent real-world situations as inequalities. benjaminpohle.com (GSO) is a free, public website providing information and resources necessary to help meet the educational needs of students.
Solve the system by creating a table.
Explain the real-world meaning of the solution, and locate the solution on a graph. Solution a. Let x represent the number of minutes, and let y represent the charge in dollars.
The charge is the monthly fee plus the rate times the number of minutes. Here is the system of equations. b.
Create a table from the equations.