Here are my notes for my differential conditions course that I educate here at Lamar University. Notwithstanding the way that these are my “class notes”, they ought to be available to anybody needing to figure out how to fathom differential conditions or requiring an update on differential conditions.
I’ve attempted to make these notes as independent as could reasonably be expected thus all the data expected to peruse them is either from a Calculus or Algebra class or contained in different areas of the notes.
Here are two or three admonitions to my understudies who might be here to get a duplicate of what occurred on a day that you missed.
Since I needed to make this a genuinely complete arrangement of notes for anybody needing to learn differential conditions have incorporated some material that I don’t, for the most part, have the opportunity to cover in class and in light of the fact that this progression from semester to semester it isn’t noted here. You should discover one of your individual colleagues to check whether there is something in these notes that wasn’t canvassed in class.
In general, I try to work problems in class that are different from my notes. Be that as it may, with Differential Equation a significant number of the issues are hard to make up spontaneously thus in this class my class work will pursue these notes genuinely close similarly as worked issues go. All things considered, I will, once in a while, work issues off the highest point of my head when I can to give a larger number of models than only those in my notes. Additionally, I frequently don’t have time in class to work the majority of the issues in the notes thus you will locate that a few areas contain issues that weren’t worked in class because of time limitations.
Sometimes questions in class will lead down paths that are not covered here. I attempt to foresee however many of the inquiries as could reasonably be expected when composing these up, yet actually I can’t envision every one of the inquiries. Sometimes a very good question gets asked in class that leads to insights that I’ve not included 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 to some degree identified with the past three things, however, it is significant enough to justify its own thing. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Utilizing these notes as a substitute for class is at risk to get you in a tough situation. As effectively noted not everything in these notes is shrouded in class and regularly material or bits of knowledge not in these notes is canvassed in class.
Here is a listing (and brief description) of the material that is in this set of notes.
Basic Concepts – In this part we present a large number of the essential ideas and definitions that are experienced in a common differential conditions course. We will likewise investigate heading fields and how they can be utilized to decide a portion of the conduct of answers for differential conditions.
Definitions – In this area a portion of the regular definitions and ideas in a differential conditions course are presented including request, straight versus nonlinear, beginning conditions, introductory worth issue and interim of legitimacy.
Direction Fields– In this segment we talk about bearing fields and how to outline them. We additionally examine how course fields can be utilized to decide some data about the answer for a differential condition without really having the arrangement.
Last Thoughts – In this area we give two or three last musings on what we will take a gander at all through this course.
First Order Differential Equations – In this section we will take a gander at a few of the standard arrangement techniques for first request differential conditions including direct, distinct, careful and Bernoulli differential conditions. We additionally investigate interims of legitimacy, harmony arrangements, and Euler’s Method. Moreover, we model some physical circumstances with first request differential conditions.
Linear Equations – In this section we solve linear first order differential equations, i.e. differential equations in the form y′+p(t)y=g(t) 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.e. differential equations in the form N(y)y′=M(x)N(y)y′=M(x). We will give a derivation of the solution process to this type of differential equation. We’ll also start looking at finding the interval of validity for the solution to a 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.e. differential equations in the form y′+p(t)y=yny′+p(t)y=yn. This section will also introduce the idea of using a substitution to help us solve differential equations.
Substitution – In this section we’ll pick up where the last section left off and take a look at a couple of other substitutions that can be used to solve some differential equations. In particular we will discuss using solutions to solve differential equations of the form y′=F(yx)y′=F(yx) and y′=G(ax+by)y′=G(ax+by).
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 exits), population 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).
Equilibrium Solutions – In this section we will define equilibrium solutions (or equilibrium points) for autonomous differential equations, y′=f(y)y′=f(y). We discuss classifying equilibrium solutions as asymptotically stable, unstable or semi-stable equilibrium solutions.
Euler’s Method – In this section we’ll take a brief look at a fairly simple method for approximating solutions to differential equations. We derive the formulas used by Euler’s Method and give a brief discussion of the errors in the approximations of the solutions.