EGZISTENCIJA I KONSTRUKCIJA NA POLINOMNO RE[ENIE NA EDNA PODKLASA LINEARNI HOMOGENI DIFERENCIJALNI RAVENKI OD VTOR RED
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1 8 MSDR 004, (33-38) Zbonik na tudovi ISBN god. COBISS.MK ID 6903 Ohid, Makedonija EGZISTENCIJA I KONSTRUKCIJA NA POLINOMNO RE[ENIE NA EDNA PODKLASA LINEARNI HOMOGENI DIFERENCIJALNI RAVENKI OD VTOR RED Elena Haxieva, Boo M. Pipeevski Elektotehni~ki fakultet Skopje e mail: boom@etf.ukim.edu.mk e mail: hadzieva@etf.ukim.edu.mk Apstakt : Vo ovoj tud se azgleduva difeencijalna avenka od vid (). So metod na tansfomacija i koistewe na soodvetni ezultati e izdvoena edna podklasa difeencijalni avenki od vid () koja ima edno polinomno e{enie za koe e konstuiana soodvetna fomula vo kone~en vid. I. Vo [] e poka`ano deka difeencijalna avenka od vid: (x-x )(x-x )(x-x 3 )z + (β x + β x + β 0 )z +(γ x + γ 0 ) z = 0, () x x x 3, x x, x,x, x 3, β, β, β 0, γ, γ 0 R. ima edno polinomno e{enie ako postoi pioden boj n (pomaliot ako postojat dva) taka {to da se zadovoleni uslovite n + (β - )n + γ = 0, β 0 + x x 3 - (x +x 3 )x + (β + )x + β x = 0, () γ 0 β + γ 0 - γ β 0 + (γ +β )( γ x +γ 0 )x = 0, Pi toa op{toto e{enie e dadeno so fomulata z = e -F {(x+k)(x-x ) n- (x-x 3 ) n- e F [C + C (x-x ) n- (x-x ) -n (x-x 3 ) -n (x+k) - dx ] } (n-), 33
2 kade {to Mx + N F =, M=β -, N=β +x + x β, ( x x )( ) x x3 x γ + n( β + xβ + xγ + γ 0 ) K = γ Vo [ ] e poka`ano deka istata avenka () ima edno polinomno e{enie ako postoi pioden boj n taka {to da se zadovoleni uslovite n + (β - )n + γ = 0, 3n() + () β + γ =0, (3) n()( - x - x - x 3 ) + () β + γ 0 =0 Pi toa op{toto e{enie e dadeno so fomulata z = AF - [A n F(C + C A -() F - dx )] (), kade {to A = (x-x )(x-x )(x-x 3 ), B = β x + β x + β 0, F = B dx A e. II. Neka e dadena difeencijalnata avenka od vid: x x, x,x, b,b 0,c 0 R. (x-x )(x-x )y + (b x+b 0 )y + c 0 y = 0, (4) Vo [3] e poka`ano deka avenkata (4) ima op{to e{enie polinom ako i samo ako postojat piodni boevi m i n taka {to se zadovoleni uslovite n( n ) + nb + c = 0, 0 (n + m ) + b = 0 ( + n) x + ( m + n ) x b 0 za nekoe {0,,,..., m -}. = 0, (5) So koistewe na uslovite (5), klasata lineani difeencijalni avenki (4) koi imaat op{to e{enie polinom go ima sledniot op{t vid: 34
3 ( x x )( x x ) y" [(n + m ) x ( + n) x ( m + n ) x ] y' + n( n + m) y = 0, (6) i op{toto e{enie }e bide dadeno so fomulata y = C P n (x) + C P m (x), odnosno d n y = C ( ) [( ) ( x x ) + x x m x x ( x x ) m+ n ] + dx d n + C ( x x ) + ( x x ) m x x x x m+ dx n [( ) ( ) ( x x ) ( x x ) m dx]. (7) So zamenata y = x z, avenkata (4) se tansfomia vo avenkata x(x-x )(x-x )z +[(b +)x +(b 0 x x )x + x x ]z + [ (b + c 0 )x +b 0 ]z = 0, ( ) odnosno avenkata x(x-x )(x-x )z + (β x + β x + β 0 )z +(γ x + γ 0 ) z = 0, ( ) Ovaa avenka pipa a na klasata difeencijalni avenki od vid () koja e izu~uvana vo tudovite [,]. Kako {to e poka`ano vo to~ka I., vo tie tudovi se dobieni pove}e gupi dovolni uslovi za egzistencija i konstukcija na edno polinomno e{enie. Neka za difeencijalnata avenka ( ) odnosno ( ) se zadovoleni uslovite (5) i neka y = P n (x) i y = P m (x) se polinomnite e{enija na avenkata (4) odnosno (6). Vo soglasnost so zamenata avenkata ( ) odnosno ( ) ima dve e{enija z = x Pn (x), z = x Pm (x), koi vo op{t slu~aj ne se polinomi. Neka fomiame lineana kombinacija λp n (x) + P m (x) i neka go opedelime λ taka {to λp n (x) + P m (x) = x Q m- (x), pi {to Q m- (x) e polinom od n + m - vi stepen. Vo soglasnost so 35
4 smenata polinomot Q m- (x) }e bide edno patikulano e{enie na avenkata ( ) odnosno ( ). So toa e doka`ana slednata teoema: TEOREMA: Neka e dadena difeencijalnata avenka ( ) odnosno ( ). Ako postojat piodni boevi m i n taka {to se zadovoleni uslovite n + ( β 3) n + γ β + = 0, n + m + β 3 = 0 (8) ( + n) x + ( m + n ) x γ = 0, 0 za nekoe {0,,,..., m -}, toga{ avenkata ( ) odnosno ( ) ima edno polinomno e{enie dadeno so fomulata z = x [λpn (x) + P m (x)], (9) kade {to polinomite P n (x) i P m (x) se dadeni so fomulite (7). Pi toa paametaot λ e opedelen so toa {to slobodniot ~len na polinomot λp n (x) + P m (x) da bide ednakov na nula. Vo soglasnost so avenkata (6), avenkata ( ) odnosno ( ), klasata avenki definiana so teoemata }e go ima op{tiot vid: x( x x )( x x ) z + {(3 n m) x + [( n + m 3) x + ( n + ) x ] x + [( n n + nm m + ) x + ( n + m ) x + ( n + ) x ] 0. + x x} z + z = III. Pime. Difeencijalnata avenka x(x-)(x-)z + (-3x + x + 4)z + 8 z = 0, gi zadovoluva uslovite od teoemata i vo soglasnost so fomulata (9) ima edno polinomno e{enie z = 3x 4 0x x 60x Op{toto e{enie }e bide dadeno so fomulata 36
5 z = C (3x 4 0x x 5x 8 60x + 30) + C. x Pime. Difeencijalnata avenka x(x-)(x-)z + (-3x + x + 4)z + (3x + 7)z = 0, gi zadovoluva uslovite od teoemata i vo soglasnost so fomulata (9) ima edno polinomno e{enie z = x 3 88x + 68x 96. Op{toto e{enie }e bide dadeno so fomulata z = C (x 3 88x 6x x + 68x 96) + C 6 + x Pime 3. Difeencijalnata avenka x(x-)(x-)z + (x + 7x - )z + (-4x + )z = 0, gi zadovoluva uslovite () i ima edno polinomno e{enie z = x + 3x + 6. Pime 4. Difeencijalnata avenka x(x-)(x-)z - (x + x + )z + (x + 8)z = 0, gi zadovoluva uslovite (3) i ima edno polinomno e{enie z = 9x - x -. Zabele{ka. Difeencijalnite avenki dadeni vo pimeite i ne gi zadovoluvaat uslovite () i (3) t.e. ne pipa a na klasite difeencijalni avenki tetiani vo tudovite [] i []. Lesno mo`e da se poka`e deka difeencijalnite avenki dadeni vo pimeite 3 i 4 koi gi zadovoluvaat uslovite () odnosno (3), ne gi zadovoluvaat uslovite od teoemata. Spoed toa mo`e da se zaklu~i deka e po{iena klasata difeencijalni avenki od vid () koi imaat edno polinomno e{enie.. 37
6 Zabele{ka. Difeencijalnite avenki od vid () koi gi zadovoluvaat uslovite od teoemata mo`at da imaat i op{to e{enie polinom ako se dodade uslovot polinomnoto e{enie P n (x) na difeencijalnata avenka (4) dadeno so soodvetnata fomula da ima sloboden ~len ednakov na nula. Zabele{ka 3. Vo [4] e poka`ano deka kompleksnite polinomi koi se e{enija na difeencijalni avenki od klasata () koi gi zadovoluvaat uslovite (), se otogonalni na ku`en lak. On existention and constuction of polynomial solution of a subclass linea homogeneous diffeential equations of second ode Elena Hadzieva, Boo M. Pipeevski Depatment of Electical Engineeing hadzieva@etf.ukim.edu.mk ; boom@etf.ukim.edu.mk Abstact: In this aticle we obseve diffeential equation of type (). By method of tansfomation and by using some pevious esults, we come to a subclass of diffeential equations of type () which has one polynomial solution. A fomula of that solution is constucted. Liteatua. Boo Pipeevski : Su une fomule de solution polynomme d une classe d equations doffeentielles lineaes du duxieme ode., Bulletin mathematique de la SDM de SRM, tome 7-8, p. 0-5, 983/84, Skopje. Boo Pipeevski : One genealization fo ones of Rodiges fomula ; Poceedings, Depatment of Electical Engineeing, tome 5 (987) p.93-98, Skopje 3. Boo Pipeevski; Su des equations diffeentielles lineaies du duxieme ode qui solution geneale est polinome, Depatment of Electical Engineeing, Poceedings N 0 4, yea 9, 3-7, Skopje, Boo Pipeevski; On complex polynomials othogonal to cicle ac, Sedmi makedonski simpozium po difeencijalni avenki, Zbonik na tudovi, st. - 6, Ohid,
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