14. starptautiskÄ konference 2012 - Latvijas JÅ«ras akadÄmija

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Proceedings of 14th International conference „Maritime Transport and Infrastructure - <strong>2012</strong>”APPLICATION OF DIFFERENTIAL CALCULUSJevgenija Kovalova, Voldemārs BarkānsLatvian Maritime academy, Flotes iela 5B, Riga, LV-1016, Latvja;Voldemars.Barkans@inbox.lvAbstractThe aim of the mathematics course of bachelor’s degree programme:- to solve practical tasks;- to develop independent learning skills;- to develop logical thinking;- to develop ability to achieve goals;- to build a mathematical model of the problem and solve it.IntroductionEvery subject contains information necessary to become a knowledgeable and functionalmember of our society. As we become more technologically dependent, technical reasoning is neededfor survival.Bachelor degree programs in technical mathematics course designed to familiarize studentswith the mathematical methods, as well to solve practical tasks in the specialty, students developindependent work skills, develop logical thinking, the ability to reach a planned target.One of the most important and most difficult tasks is to learn the skills to expressmathematical language. Differential equations play a prominent role in engineering, physics,economics, and other disciplines, which in fact are used to solve these equations. Some illustrativeexamples will be shown in this project.Problem 1. Parabolic mirror. Establish, what should be the shape of the searchlight mirror inorder for a pointed source of light to be reflected as a beam of parallel rays.M(x;y)Ny yxyStarsP 2xxFig. 1 Parabolic mirrorSolution. Due to its symmetry, mirror has a shape of rotation surface. We will see that themirror has a shape of rotation paraboloid.Let’s assume that initial point of coordinates is the same as source of light, but the Ox axisdirection of abscises is the same as direction of beam of rays.We look at assumed section line of surface y = y(x) in the plane XOY (Fig.1) and freelychoose a point on this line M(x;y). At the point M we mark a tangent PM for the line y = y(x). AnglesPMO and KMO are identical, as angles of falling ray OM and reflected ray. Angle MPO is angle ofcurve tangent PM direction: k tg yx . All three of these angles are equal (Fig.1) and 2. 118
Proceedings of 14th International conference „Maritime Transport and Infrastructure - <strong>2012</strong>”Thereforeor (Fig. 1)2tg2ytg tg 2 221tg 1y(1)ytg .x(2)We arrive at first order homogenous (nonlinear) differential equation:2yy21yx2 2xy y1 y(3)From the equation (3) we get:2 x x yy y(4)and introduce new unknown function:yu ; y ux;y ux u , wherexduu (5)dxAfter introduction of new variables in the equation (4), we separate variable and integrate:ududx ln C22 (6)1u 1u 1xFor the integral at the left side of equation we make substation of integration variable:2 21 u t ; 2udu 2tdt(7)and by integration we arrive at: ln t 1 ln x ln C . (8)Applying formulas (5) and (7), from (8) we get: y 1 1 x 2Cx. (9)After removing square root and making changes in constant C, we arrive at parabolic equation:y22 2Cx C . (10)When parabola (10) rotates around Ox axis a surface is created, which is called a rotation paraboloid.Problem 2. Radioactivity. The disintegration speed of radium is proportional to quantity ofsubstance Q. We know that after time of t 1 only half of the initial quantity Q 0 is left. Calculate amountof radioactive radium left after time of t .10t 1119
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14. starptautiskÄ konference 2012 - Latvijas JÅ«ras akadÄmija

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