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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.Roots of the characteristic polynomial. An order-d homogeneous linear recurrence with constant coefficients is an equation of the form = − + − + ⋯ + −, where the d coefficients c i (for all i) are constants, and ≠..
A constant-recursive sequence is a sequence satisfying a recurrence of this form. There are d degrees of freedom for solutions to this recurrence, i.e., the initial.
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Edit Article How to Factor a Cubic Polynomial. In this Article: Article Summary Factoring By Grouping Factoring Using the Free Term Community Q&A This is an article about how to factorize a 3 rd degree polynomial. We will explore how to factor using grouping as well as using the factors of the free term.
A polynomial function has real coefficients, a leading coefficient of 1, and the zeros -1, -2, and 5. Write a polynomial function of least degree in .
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Write a polynomial with rational coefficients having roots 3, 3 + i, and 3 - i.? +1 vote Part I: Write the factors (in the form x - a) that are .