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Lecture_1(Ordinary Differential Equations)

by: Heman

Lecture_1(Ordinary Differential Equations) ME-5331

Marketplace > University of Texas at Arlington > Aerospace Engineering > ME-5331 > Lecture_1 Ordinary Differential Equations
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Ordinary Differential Equations
engineering analysis
Prof Dora
Class Notes
Mathematics, Differential Equations, Ordinary, Engineering, graduate




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This 8 page Class Notes was uploaded by Heman on Thursday April 21, 2016. The Class Notes belongs to ME-5331 at University of Texas at Arlington taught by Prof Dora in Fall 2015. Since its upload, it has received 27 views. For similar materials see engineering analysis in Aerospace Engineering at University of Texas at Arlington.


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Date Created: 04/21/16
ME5331  Lecture 1 Ordinary Differential Equations (ODEs) 1. Introduction to Differential Equations An ordinary differential equation (ODE) is a mathematical equality involving a function and its derivatives. The simplest type of differential equation is where is a given function of and is the derivative ofwith respect to . The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. We can also say that the derivative of a function as its rate of change. The simple derivative of a function with respect to a variable is denoted either or . When derivatives are taken with respect to time, we often denote them using Newton’s dot notation for fluxions, When a derivative is taken times, we use the notation or In mechanics we typically use Newton’s fluxion notation etc. We define the derivative of a function with respect to the variable as but may also be calculated more symmetrically as provided the derivative is known to exist. An ODE is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function. 2. Definitions ORDER  The order of a differential equation is the order of the highest derivative it contains. An ODE of order is an equation of the form where is a function of , is the first derivative with respect to , and is the th derivative with respect to . Examples: Dr. Dora Musielak Page 2 SOLUTION  A solution of an ODE on the interval is a function such that exist for all and for all A function is a solution of a differential equation, over a particular domain of the independent variable, it its substitution into the given equation reduces it to an identity valid everywhere within that domain. Example: Verify that the function is a solution of the differential equation on the open interval Usually the unknown function is subjected to conditions at one or more point on the interval under consideration: INITIAL-VALUE PROBLEM (IVP)  a differential equation together with the initial conditions is called an initial-value problem. Initial conditions are conditions specified at a single point, usually the end point of the interval. The independent variable is typically the time (not always but often). BOUNDARY-VALUE PROBLEM (BVP)  a differential equation together with boundary conditions is called a boundary-value problem. Boundary conditions are conditions specified at both ends of the interval. The independent variable is typically a space variable. Example 1: Example 2: Example 3: Solution: Note that the function is continuous in the interval but its antiderivative is not an elementary function. Remember, an antiderivative of a function is a differentiable function whose derivative is equal to , i.e., Let us use as a dummy variable of integration, so we integrate the given diff eq., Dr. Dora Musielak Page 3 Applying the IC, , we obtain the solution to the IVP: In general, for an IVP we seek a function defined on some interval I containing that satisfies the differential equation and the initial conditions specified at . In the case of a second order IVP, a solution curve must pass through the point and have slope at this point. Linear and Non-Linear Differential Equations. A linear ordinary differential equation of order , in the dependent variable and the independent variable , is an equation that is in, or can be expressed in, the form where is not identically zero. Observe that: (1) the dependent variable and its various derivatives occur to the first degree only; (2) no products of and/or any of its derivatives are present; and (3) no transcendental functions and/or its derivatives occur. In general, a Linear equation is said to be linear if it is linear in the “unknowns” , , , etc. Also note that the functions , . . . , , need not be linear functions of . A nonlinear ordinary equation is an ODE that is not linear. The following two examples should convey the general idea of linearity. Example 1: is a 2nd order, linear differential equation. Dr. Dora Musielak Page 4 Example 2: is a non-linear, ordinary differential equation of order 3. The equation is non-linear because of the presence of the term which is a quadratic function of the unknown function . Linear ODEs are further classified according to the nature of the coefficients of the dependent variables and their derivatives. For example is linear with constant coefficients, while the equation is linear with variable coefficients. Homogeneous and Nonhomogeneous Differential Equations The ordinary, nth order, differential equation of the form is a nonhomogeneous equation if However, if then it is a homogeneous equation. How differential equations originate? Where do they come from? Origin and Application of Differential Equations Differential equations represent natural and scientific processes and engineering system behaviors. Differential equations are found in connection with a myriad of problems such as  Determining the motion of objects, projectile, rocket, satellite, planet, ...  Conduction of heat in a rod or slab  Determining the charge or current in an electric circuit  Vibrations of a wire, membrane, beam, ... Dr. Dora Musielak Page 5  Rate of decomposition of radioactive substances  Rate of growth of a population  Thermo-chemical reaction of chemicals  Determination of curves that have specified geometrical properties  and so on The mathematical formulation of those and millions of other problems give rise to differential equations. In the examples above, you know that the items involved obey certain scientific laws. These laws involve rates of change of one or more quantities with respect to other quantities, e.g., , and so on. In other words, rates of change are expressed mathematically by derivatives. The scientific laws themselves become mathematical equations involving derivatives, that is, differential equations. For example, This equation is the basis of Newtonian mechanics. In the process of mathematical formulation, certain simplifying assumptions generally have to be made in order that the resulting differential equations be solvable. If the actual phenomenon or situation in a certain aspect of the problem is too complicated, we may modify it by assuming an approximate situation that is simpler, yet it still maintains the nature of the problem. Many times the resulting differential equation is actually that of an idealized situation. However, we still obtain from that equation a great deal of knowledge and thus it is very valuable. To obtain useful information from a differential equation, we solve it! Solution Techniques The purpose of this course is to learn some basic techniques for solving differential equations and to study the general properties of the solutions of differential equations. Be aware that we rarely find the solutions of a differential equation by systematically manipulating an equation until the unknown function is isolated. More often we find the solutions by first constructing, or even guessing, some solutions and then applying some general theorems to verify that every solution can be found in this manner. Indeed, there exists no general algorithm for solving differential equations. However, if one knows how to construct a solution for one differential equation, one can often generalize the technique to a whole class of differential equations. For this reason the first step in solving a differential equation is to identify its general class. 3. Mathematical Modeling Modeling is the process of writing a differential equation to describe a physical situation. The differential equations that you will use in your profession are there because somebody, at some time, modeled a situation to come up with the differential equation that you are using. Dr. Dora Musielak Page 6 You cannot learn in one course how to go about modeling all physical situations. However, you must learn some aspects of the modeling process and become familiar with what is involved in modeling. Keep in mind that, to model a physical process, we must make assumptions that do not accurately depict reality in most cases, but without them the problems would be intractable. The modeling process For more on mathematical models, please read the introduction to modeling, Section 1.3 in the textbook. Basic ODEs in Engineering Newton’s Second Law of Motion Governing equation of a Mass-spring System is linear displacement Circuit equation is electrical current with applied voltage of strength Pendulum equation determines angular motion Dr. Dora Musielak Page 7 Population growth/decay equation Hanging cable equation is deflection of cable Loaded beam equation is deflection of beam Riccati equation D’Alembert-Lagrange equation Clairaut equation Euler-Cauchy equation Bernoulli equations Dr. Dora Musielak Page 8


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