Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. Included are most of the standard. and their solutions. In studying integration, you already have considered solutions to very simple differential equations. For example, when you look to solving. A differential equation is a mathematical equation that relates some function with its derivatives. Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his work "Methodus.
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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 differential equation calculus the solution.
Final Thoughts — In this section we give a couple of final thoughts differential equation calculus 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 differential equation calculus 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 differential equation calculus 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 equation calculus 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 differential equation calculus case that will not involve complex differential equation calculus.
We will use reduction of order to derive the second solution needed to get a general solution in this case. Reduction differential equation calculus Order — In this section we will discuss reduction of order, the process used to derive the solution to the repeated roots case for homogeneous linear second order differential equations, in greater detail.
This will be one of the few times in this chapter that non-constant coefficient differential equation will be looked at.
Fundamental Sets of Solutions — In this section we will a look at some of the theory behind the solution to second order differential equations. We define fundamental sets of solutions and discuss how they can be used to get a general solution to a homogeneous second order differential equation.
We will also define differential equation calculus Wronskian and show how differential equation calculus can be used to determine if a pair of solutions are a fundamental set of solutions.
More on the Wronskian — In this section we will examine how the Wronskian, introduced in the previous section, can be used to determine if two functions are linearly independent or linearly dependent.