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Section 16-5. Differential Equations: Probability The first half of the course will be devoted to solving differential equations. We will learn how to use the mean value theorem; we will solve second order linear differential equations, solve systems of first order linear differential equations that are underdetermined, and transform ordinary differential equations into differential equations in an unknown angle variable. We will also consider boundary value problems with irregular singularities and regular singularities. We will study the absence of a determining factor in solving differential equations. For each problem we will attempt a detailed solution, using a combination of direct and graphical methods, for a small selection of special cases to test our understanding. For each problem we would complete a long written solution, based on our own work. The final project is to write a computer program to solve a special class of differential equations, the Frobenius method. The expected duration of this project is approximately two weeks. The project will be completed on the first day of class.
As an area of mathematics, analysis is concerned with problems which are inherently of a finite nature, e.g., problems in geometry, number theory, and analytic number theory.Geometric probabilities and integrals are used to deal with a wide variety of phenomena, such as coin flipping, random walks (Brownian motion), zeros of polynomials, uniform distributions, and so on. Probability theory deals with events and their mathematical relationships; these topics cover a wide variety of fields, from handicapping college football teams to neuron firing in a brain. Equations of mathematical physics may be solved analytically, and are often solved numerically (i.e., using computers instead of analytical methods). There are also many physical problems which may be formulated mathematically as differential equations. These include relativity and the related field of elementary particle physics, as well as fluid mechanics and the related field of stellar mechanics. The latter is the field of dust-fluid interaction, dust assemblies from a micron to kilometer size. This course, as well as the following two courses, will include the working out of elementary physical models and the study of mathematical problems in their imaginary or physical form. The main content of this course will be statistical mechanics, an area of physics dealing with macroscopic systems in thermodynamics, molecular dynamics, solid state physics and surface science. d2c66b5586
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