ON QUASI-CHARACTERISTIC EQUATIONS AND SYSTEMS FOR REAL DIFFERENTIAL EQUATION

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1 ON QUASI-CHARACTERISTIC EQUATIONS AND SYSTEMS FOR REAL DIFFERENTIAL EQUATION Jelena Vujaković *, Miloje Rajović ** Abstract. In linear homogenous differential equation of second order with constant coefficients, implementation of substitution y = ep( r), where r is also constant, we obtain algebraic characteristic equation, which not only determines the type of solution-monotonous or oscillatory, but at the same time gives the solution itself. In this paper we shall prove that analogous principle applies even when coefficients in this equation are arbitrary continuously-differentiable functions. We will show that characteristic equation can also be implemented for non-constant coefficients and that the systems of those and quasi-characteristic equations are possible. This can be obtained by implementation of comple variable. Keywords. Differential equation, a quasi-characteristic equation, a seriesiteration method, Sturm's functions Introduction and preliminaries According to Liouville's principle it has been known that in the solutions of linear homogenous differential equation of second order the eponential functions ep( r ), ep( ir ) are important. They can also be complicated and they give two general solution types: monotonous and oscillatory. Furthermore, in solutions are very frequent and product of type y = uv, where one multiplier is eponential function and other is some arbitrary function, but the point is that the both multiplies depend on constant coefficients. For linear homogenous differential equations of second order y + a y + b y = (.) with non-constant coefficients a( ) and form y f ig where f = f ( a, b ) and g g( a, b ) 39 b we could find solution in comple = ep (.) =. Substituting (.) and their derivates in equation (.) and then separating real and imaginary part we obtain g, which we shall call system of system with respect to unknown f ( ) and quasi-characteristic equations for (.). It is clear that the following lemma is valid

2 4 J. Vujakovic, M. Rajovic /God. Zb. Inst. Mat. 4 (3) Lemma.. For linear homogenous differential equations of second order D =,, with minor (.) where a( ) and b( ) are positive coefficients on [ ) increase ( b < ep a d ), by substitution y ep( r) ( ) = we obtain slowly increasing monotonous solution y= ep( r) = c ep a( d ) c b ep a( ddd ) (.3) where c and c are arbitrary integral constants. Proof. Suppose that the solution of equation (.) is y = ep( r). (.4) Finding here the first and the second derivate and substituting them in (.), we obtain eact equation ep( r) r ( r ) a r b =. However, if we are looking for solutions in which the ep( r ) is not high, that is r eists but is not big; namely, let r <, then eact equation (.4) is transform on quasi-characteristic equation r + a r + b = (.5) where a( ) and substitution r = v in (.5) we obtain linear homogeneous differential b are positive coefficients with minor increase. Further, by equation of first order out which we find solution for v( ) and afterwards for r( ) According to the (.4) from (.6) we find approimate solution (.3) of equation (.), which gives slowly increasing monotonous solution. The main result In this section we state and prove our main theorems of this paper... Oscillatory solutions with equidistant zeros. Theorem.. Linear homogenous differential equation of second order (.), a and with non-constant coefficients a( ) < and b( ), under conditions b( ) are continuous function in [, ) and a b const =, (.) 4 has oscillatory solution with equidistant zeros. The general solution is in form

3 Ј. Вујаковиќ, М. Рајовиќ/ Год. Зб. Инсt. Мат. 4 (3) y = ep a d ( ccos k + csin k), k = const. (.) Proof. In order to solve (.) it is best to apply substitution y = f ep ( ik), k = const (.3) By finding the derivative and by substitution in (.) we obtain ep( ik) f k f + a f + b f + i( kf + ka f ) =. Hence, since ep( ik), by separating real and imaginary part we obtain system of quasi-characteristic equations f k f + a f + b f =. (.4) kf + ka f = >From second equation of system (.4) we find f ep = a d. When this f is substituted in first equation of (.4) together with derivates f and f, after brief calculation, we obtain (.), that is a b = k = const, k >. 4 If this k is substituted in (.3) we have general solution a a y = ep a d ep i b ± (.5) 4 which is oscillatory, but with equidistant zeros. a It is remarkable that if epression Φ = b in (.) is not 4 constant, then in (.) appears function a cos b = cos( Φ ) 4. (.7) a sin b = sin ( Φ ) 4 That is how we have proven net corollary Corollary.. Linear homogenous differential equation of second order (.), where a( ) and b( ) are continuous function on [, ) and b( ) applie correlation (.) has oscillatory solutions with equidistant zeros. However, if k is not constant but k = Φ is certain function of, from (.7) we can see that in that case solutions are also oscillatory, but depend on complicated function b a Φ =. 4

4 4 J. Vujakovic, M. Rajovic /God. Zb. Inst. Mat. 4 (3) Beside that, in a slightly different way, that is, by the help of the system of quasi-characteristic equations, we proved here an important theorem for differential equations of the second order. Theorem.3. (Transformation to canonic form). Linear homogeneous differential equation of second order (.), for continuous coefficients a( ) and b( ) on D = [, ), by substitute y z ep = a d transforms on canonic form of equation z +Φ z =, (.8) Φ =Φ a, b is given with (.6). where ( ).. Sturm's function. The theorem.3. about transformation of linear homogeneous differential equation of second order (.) on canonic equation (.8) is very suitable for solution through iteration. >From integral form of equation (.8) z = c + c Φ z d for selection of constants ( c, c ) = (,) and (, ) (,) [ ] n [ n z ] = Φ z d [ n] [ n ] z = Φ z d c c = follows iterations (.9). (.) In papers []-[4] is shown that iterations (.9) and (.) converge for continuous functions a( ) and b( ) which satisfy Lipschitz's condition on area D = [, ). Then sequences (.9) and (.) determine continuous oscillatory functions which we called Sturm s functions. Hence, it is obtained that equation (.8) has solutions k = ( ) Φ Φ = Φ z d d cos, k = k -double integrals k = + ( ) Φ Φ = Φ z d d sin, k = k -double integrals for positive Φ which is given by (.6). Net, for Φ = const we obtain ordinary Euclidean cosine and sine, and for other variable functions Φ, as solutions are obtained function very close and similar to ordinary cos λ and sin λ, but with non-equidistant zeros, subjected to classic Sturm s theorems.

5 Ј. Вујаковиќ, М. Рајовиќ/ Год. Зб. Инсt. Мат. 4 (3) Sturm's functions (.9) and (.) can also be approimately epressed by ordinary cosine and sine, but from complicated function, so-called frequency Φ, that is function sin cos cos Φ, sin Φ Φ ( Φ ) Φ Zeros of cosine and sine solution are in solutions of equation π Φ = ( k ), k =,,. Φ = nπ, n =,,, Beside that, Prodi's theorem (see [6]) also will be applied: Function sin Φ when +, while function cos ( ) is limited when +. Φ In oposite case, when Φ is negative, canonic equation has solutions z Φ z = = Φ Φ = Φ z d d cos, k = k -double integrals = + Φ Φ = Φ z d d sin, k = k -double integrals which for Φ = give ordinary cosh and sinh. It also can be proven by Sturm's theorems []-[5] that they can have at the most one zero. That implies a new general theorem Theorem.4. (The general theorem). Linear homogeneous differential equation of second order (.), for continuous coefficients a( ) and b( ) on D = [, ) has solutions in the form of Sturm's functions y = ep a d ( csin ccos ) +, for Φ ( ) > Φ Φ or y ep ( sinh cosh ) a d = c c +, for Φ ( ) < Φ Φ. It can be shown that certain classes of discontinuous Sturm's functions are possible, which makes these results very important..3. General oscillatory solutions For oscillatory solutions of equation of second order (.), are very useful characteristic particular integrals in form y = f cos g, y = f sin g..

6 44 J. Vujakovic, M. Rajovic /God. Zb. Inst. Mat. 4 (3) It seemed to us that it is fully sensible to search for solution of equation (.) in comple form (.). We are of opinion that the net theorem could be the basic for non-linear oscillations study. Theorem.5. Equation (.) can also define non-linear oscillations of the form (.) in the case that g are determined by f and ( ) 3 f ( ) f + a f + b f c ep a d =, (.7) g = c a d d + c ep f. (.8) Proof. Suppose that the solution of equation (.) is y = f ep ( ig ). Finding here the first and the second derivate and substituting them in (.) and then by separating real and imaginary part, we obtain quasi-characteristic system of equations f + a f + f b ( g ) = (.9) f g + a f + f g =. If in second equation of this system separate functions f ( ) and g ( ) we have g f = a. From here we obtain the first integral g f c g = ep a d f, respectively the second integral (.8). From the first equation of system (.9) we found function f ( ), that is c f + a f + b f ep 3 ( a d ) = f. This is equation of the second order, which also can have oscillatory solution. ACKNOWLEDGMENT The authors are grateful to the Ministry of Science and Environmental Protection of Serbia. REFERENCES [] M. Lekić, Sturm theorem through iteration, Dissertation, University Kosovska Mitrovica, Faculty of Sciences and Mathematics, 7 ( in Serbian ) [] D. Dimitrovski, M. Mijatović, A Series-Iteration Method in the Theory of Ordinary Differential Equations, Hadronic Press, Inc. Florida, USA, 997 [3] D. Dimitrovski, M. Rajović, R. Stojiljković, On type, forme and supremum of the solution of the linear differential equation of the second order with

7 Ј. Вујаковиќ, М. Рајовиќ/ Год. Зб. Инсt. Мат. 4 (3) entire coefficients, Applicable Analysis and Discrete Mathemtics, Vol. I, No, (7), 36-37, (Symposium MAGT, ETF, Belgrade, Serbia, 6.) [4] M. Rajović, R. Stojiljković, D. Dimitrovski, Transformation of linear nonhomogeneus differential equations of the second order to homogeneous, Computers and Mathematics with Applications 57 (9), 64-6 [5] W. O. Amrein, A. M. Hinz, D. B. Pearson, Sturm-Liouville Theory, Past and Present, Birkhäuser Verlag, Basel, Switzerland, 5 [6] G. Prodi, Un osservazzione sugli intergale dell equazione y + A y = nel caso A + per +, Rend. Acad. Lincei (8), 8 (95),

8 46 J. Vujakovic, M. Rajovic /God. Zb. Inst. Mat. 4 (3) ЗА КВАЗИ-КАРАКТРЕИСТИЧНИТЕ РАВЕНКИ И СИСТЕМИ ЗА РЕАЛНИ ДИФЕРЕНЦИЈЛАНИ РАВЕНКИ Јелена Вујаковиќ, Милоје Рајовиќ Апстракт. Кај линеарната хомогена диференцијална равенка од втор ред со константни коефициенти, примената на замената y = ep( r), каде r е исто така константа, добиваме алгебарска карактеристична равенка која што не само го определува типот на решението - монотони или осцилаторно, но го дава и самото решение. Во овој труд ќе докажеме дека аналоген принцип може да се примени дури и кога коефициентите се произволни непрекинато-диференцијабилни функции. Ќе докажеме дека карактеристичната равенка може да се примени и за неконстатнтни коефициенти и дека системите од таква и квази-карактеристична равенка се можни. Тоа може да се направи со примена на комплексна променлива. * Faculty of Sciences and Mathematics, Lole Ribara 9, 38, Kosovska Mitrovica, Serbia, enav@ptt.rs ** Faculty of Mechanical Engineering, Dositejeva 9, 36 Kraljevo, Serbia, rajovic.m@mfkv.kg.ac.rs