ABOUT THE ZEROS AND THE OSCILLATORY CHARACTER OF THE SOLUTION OF ONE AREOLAR EQUATION OF SECOND ORDER WITH ANALYTIC COEFFICIENT

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1 Математички Билтен ISSN X (pint) Vol. 40(LXVI) No. 1 ISSN (online) 016 (73-78) UDC: Скопје, Македонија ABOUT THE ZEROS AND THE OSCILLATORY CHARACTER OF THE SOLUTION OF ONE AREOLAR EQUATION OF SECOND ORDER WITH ANALYTIC COEFFICIENT Slagjana Bsakoska Abstact In the pape, one linea aeola equation of second ode with analytic coefficients is consideed, egading the eoes and the oscillatoy chaacte of its geneal solution. In this equation the fist deivative is missing. Some theoems will be poven and some eamples will be given fo diffeent cases of the coefficient. 1. INTRODUCTION The notion of the tem comple numbe, comple vaiable and comple function f() is a few centuies old and moe than a centuy old is the idea fo epanding the opeations deivative and integal to a function of compleconjugated vaiable, iy. In 1909, G.V. Kolosov [1], duing his effots to solve a poblem fom the theoy of elasticity, has intoduced the epessions 1 ˆ [ u v i ( v u) dw y y and (1) 1 ˆ [ u v i ( v u) dw () y y known as opeatoy deivatives of a comple function W W( ) u(, y) iv(, y) fom a comple vaiable iy and iy, espectively. The opeato ules fo these deivatives ae given in the monogaph of Г. Н. Положиǔ [] (pages 18-31). In the mentioned monogaph, ae also defined the so called opeatoy integals 010 Mathematics Subject Classification. 34M45, 35Q74. Key wods and phases. aeola deivative, aeola equation, solution, eos of solution, oscillatoy chaacte, analytic function.

2 74 Slagjana Bsakoska f ( ) and f ( ) by iy and iy, espectively, fom the comple function f f () in the aea D, whee thei opeatoy ules ae poven as well, page REASONS FOR INTRODUCING THIS EQUATION AND FORMULATION OF THE PROBLEM In the theoy of eal functions a big ole has the tem oscillatoy and especially the tem peiodical, as a diect consequence of the Newton's laws. k 0 The equation d is one of the oldest diffeential equations and at the dt m same time the equation of oscillatoy pocesses (fom stetching the sping pendulum, to otating motion of bodies bounded by mutual action of gavitational foces). In the case when k i.e. m is vaiable, and if we intoduce hee a geneal function y ( ) and a egulato of the appeaance a ( ) we get a diffeential equation which is called an equation of oscillations, if 1. a ( ) 0 and y a( ) y 0 (3). a ( ) is big enough to cause oscillations, which is epessed analytically with the condition the integal a( ) d to be divegent. 0 Analogous to the equation (3) fo the functions of two comple vaiables W W(, ), would be the equation with aeola deivatives fom second ode whee, dw ˆ A(, ) W 0 (4) A is a given function and W, u, y iv, y is an unknown function by the vaiables and, which is a subject of analysis in ˆ ˆ ˆ this pape. Hee, the deivative d W d ( dw ), and the deivative ˆdW is defined with (). One of the questions aised hee is the following: Is thee an analogy with eal oscillations and whethe (4) can be called aeola equation of oscillations? Whethe the solutions of the equation (4) have eos and what is thei natue?

3 About the eos and the oscillatoy chaacte MAIN RESULT In [8], we tied to answe the simplest case of the equation (4), which is also the closest to the eal oscillations, i.e. A(, ) K i whee K is a comple constant. Thee, we consideed an aeola equation with constant coefficients dw ˆ ( i) W 0 (5) and some theoems wee poven. Also, we have consideed some eamples fo vaious cases of the coefficient K. In this pape we ae consideing the equation (4), whee A () is an analytic da function. Because of that fact, we have that ˆ ( ) 0. So, in the equation dw ˆ A( ) W 0 (6) we can conside A () as it is in the ank of "a genealied constant" in opeatoy vocabulay. So we can do the pocedues that ae maid in [8], also hee to the equation (6). We ae looking fo a paticula solution fom the following fom W e (7) whee ( A( )) and the solutions of the comple chaacteistic equation ae depending fom y,, i.e. A( ) 0 (8) 1/ ( 1/ (, y) i1/ (, y)) (9) Repeating the pocedue as in [8], i.e. using some fomulas fom comple i analysis and tigonomety, i.e. if we put ( i) e, whee and actg, fo (9) we have i.e. and actg k actg k 1/ [cos isin ], k 0,1 4 actg actg 1 i [cos sin ] 4 actg actg i [cos sin ].

4 76 Slagjana Bsakoska If we use some of the tigonometic fomulas that ae useful hee, with shot tansfomations we get and (, y) (, y) (, y) (, y) (, y) (, y) 1(, ) i (10) y (, y) (, y) (, y) (, y) (, y) (, y) (, ) (11) y i and accoding to this, the geneal solution of (6) will be: We have poven the following 1( ) ( ) 1 W(, ) C ( ) e C ( ) e. (1) Theoem. The aeola equation (6) with analytic coefficient solution (1), whee the comple functions (10) and (11). Lets see some eamples. A has geneal 1 and ae given with Eample 1. Lets conside the equation gives us the equation so, dw ˆ W 0. The substitute W e 0 whee fom we get that 1/ ( iy) 1 ( iy)( iy) y W1 e e e W e e e and now the geneal solution is ( iy)( iy) ( y ) 1 e y 1 1 W(, ) C ( ) e C ( ) e C ( ) e C ( ) e. We can see that the eoes can be only the common eoes of the "constants" C 1 () and C (). Eample. Fo the equation dw ˆ 0 whee fom we get 1/ 1 i W 0 the chaacteistic equation is i i i i 1 1 W(, ) C ( ) e C ( ) e C ( ) e C ( ) e

5 About the eos and the oscillatoy chaacte 77 o W cos( y )[ C1 ( ) C( )] isin( y )[ C1( ) C( )] It is obvious that the eoes of W can be only the common eoes of C 1 () and C (). Eample 3. Lets conside a coefficient A () without eoes, fo eample A() e W e we get So, we have now:. Then, we have the equation dw ˆ e e whee fom 1/ ie ie ie W1e, W e. W 0 and with the substitute e [ ycos ysin y] W1 e {cos[ e ( cos y ysin y)] isin[ e ( cos y ysin y)]} e [ ycos y sin y] W e {cos[ e ( cos y ysin y)] isin[ e ( cos y ysin y)]} Zeoes fo W 1 we will get fo cos 0 and sin 0 fo e ( cos y ysin y), and the eoes fo W ae the same. Fom cos 0, we get (k 1), and fom sin 0, we get n. Since this functions ae neve equal, we have that W(, ) has no eoes, ecept fo W 0, i.e. fo C1( ) C( ) CONCLUSION In [8] we concluded that in the linea aeola equation fom II ode (5), the oscillatoity eists in the solution both in the eal and in the imaginay pat and W(, ) 0 has eoes, whee the signs of and in the coefficient K i does not has any influence on that oscillatoity. Fom the pevious eamples, we can conclude that fo the consideed aeola equation fom II ode (6), eoes of its solutions can be only the common eoes of the genealied constants C 1 () and C () which is diffeent fom the Theoem in [8]. We can conclude that in dw ˆ KW 0 if K is not a constant, but it is a "genealied constant" A (), then the fomulated theoem in [8] no longe stands.

6 78 Slagjana Bsakoska Refeences [1] Г. В. Колосов, Об одном приложении теории функции комплесного переменного к плоское задаче математическои упругости, 1909 [] Г. Н. Положии, Обопштение теории аналитических фукции комплесного переменного, Издателство Киевского Университета, 1965 [3] S. B. Bank, I. Laine, On the oscillation theoy of f A( ) f 0, whee A () is entie. Tans.Ame.Math.Soc., 73, (198). No1, [4] Б. Илиевски, Линеарни ареоларни равенки (Контурна интеграција. Специјални функции од две комплексни променливи. Ареоларни Лапласови трансформации), Докторска дисертација, Скопје,199 [5] Д. Димитровски, Б. Илиевски, С. Брсакоска и други: Равенка Векуа со аналитички коефициенти, Специјални изданија на Институтот за математика при ПМФ на Универзитетот Св. Кирил и Методиј - Скопје, 1997 год. [6] D. Dimitovski, M. Rajović, R. Stoiljković, The genealiation of the I. N. Vecua equation with the analytic coefficients, Filomat No. 11 (1997), 9-3 (MR ) [7] M. Rajović, D. Dimitovski, R. Stojiljković, Elemental solution of Vecua equation with analytic coefficients, Bul. Ştiinţ. Univ. Politeh. Timişoaa Se. Mat. Fi. 41(55), no. 1 (1996), 14-1 MR [8] S. Bsakoska, About the eos and the oscillatoy chaacte of the solution of one aeola equation of second ode with constant coefficient, Mat. Bilten. Vol. 40, No. 1 (016), 55 6 Ss. Cyil and Methodius Univesity Faculty of Natual Sciences and Mathematics, Ahimedova 3, 1000 Skopje, R. Macedonia sbsakoska@gmail.com