Overview[ edit ] Definitions of complexity often depend on the concept of a confidential " system " — a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime. Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time. Warren Weaver posited in two forms of complexity:
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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.
Separable Equations — In this section we solve separable first order differential equations, i. We will give a derivation of the solution process to this type of differential equation. Exact Equations — In this section we will discuss identifying and solving exact differential equations.
We will develop of a test that can be used to identify exact differential equations and give a detailed explanation of the solution process. We will also do a few more interval of validity problems here as well. Bernoulli Differential Equations — In this section we solve Bernoulli differential equations, i.
This section will also introduce the idea of using a substitution to help us solve differential equations. Intervals of Validity — In this section we will give an in depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations.
Modeling with First Order Differential Equations — In this section we will use first order differential equations to model physical situations. In particular we will look at mixing problems modeling the amount of a substance dissolved in a liquid and liquid both enters and exitspopulation problems modeling a population under a variety of situations in which the population can enter or exit and falling objects modeling the velocity of a falling object under the influence of both gravity and air resistance.
We discuss classifying equilibrium solutions as asymptotically stable, unstable or semi-stable equilibrium solutions.
Second Order Differential Equations - In this chapter we will start looking at second order differential equations. We will concentrate mostly on constant coefficient second order differential equations. We will derive the solutions for homogeneous differential equations and we will use the methods of undetermined coefficients and variation of parameters to solve non homogeneous differential equations.
In addition, we will discuss reduction of order, fundamentals of sets of solutions, Wronskian and mechanical vibrations. We derive the characteristic polynomial and discuss how the Principle of Superposition is used to get the general solution. We will also derive from the complex roots the standard solution that is typically used in this case that will not involve complex numbers.For this problem they had to create a model, write an equation and solve the problem.
In doing this they were able to demonstrate whether or not they had gained a true understanding of how to use modeling as a tool and whether or not they understand the relationship between concrete modeling and equation solving.
Free Online Scientific Notation Calculator. Solve advanced problems in Physics, Mathematics and Engineering. Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. Aug 19, · Best Answer: The independent variable is the number of workers (which we will call 'W') and the dependent variable is the number of radios built (which we will call 'R').
From the data, the relationship is strongly linear, in fact exactly the equation Status: Resolved. Table — Chunk fields; Length: A four-byte unsigned integer giving the number of bytes in the chunk's data field. The length counts only the data field, not itself, the chunk type, or the CRC.
Zero is a . SOLUTION: Directions: Define the variables and write an equation to model the relationship in each table. (This is one table with 2 coulumns) Number of sales Total earnings 5 Algebra -> Expressions-with-variables -> SOLUTION: Directions: Define the variables and write an equation to model the relationship in each table.
Have students copy the table, write the equation and graph the ordered pairs on the Linear Relations Using Tables, Equations and Graphs worksheet. You may choose not to press for the remaining Y values asking students to substitute the given value for X and find the value for the Y column.