Construction of the solutions of non-linear high order differential equations

A C T A T E C H N I C A ỔSAV 1980 No C O N S T R U C T I O N O F T H E S O L U T IO N S O F N O N - L I N E A R H IG H O R D E R D IF F E R E N T IA L E Q U A T IO N S K im C hi - V an D ao In m any te c h n ic a l p ro b le m s we have m et the d iffe re n tia l e q u a tio n s o f second o rd er R e c e n tly , in so m e fields o f P h y sics, M e c h a n ic s a n d B io lo g y one d e a ls w ith the n o n ­ lin e a r d iffe re n tia l eq u atio n s o f th ird a n d fo u rth o rd e rs [ - ] I n th is a rt ic le the K r y lo v -B o g o liu b o v -M it r o p o ls k i [ — 8] a sy m p to tic m eth o d has been used to co n stru ct the s o lu tio n s o f n o n -lin e a r h ig h o rd e r d iffe re n tia l e q u atio n s T h e first p a g p h is co n ce rn e d w ith the a u to n o m o u s e q u a tio n a n d the second o ne — w ith the n o n -a u to n o m o u s e q u atio n S O L U T IO N O F A U T O N O M O U S E Q U A T IO N L e t us c o n s id e r the h ig h o rd e r n o n -lin e a r a u to n o m o u s d iffe re n tia l e q u a tio n o f fo rm (1.1) x iS) + a j.x ^ " + + aN- ị X + aA = e F(x, X , x w , c ) , x here e is a s mall p aram eter, 0L are real co n stants, F(x, X, ị e) is known function h a v in g e n o u g h n um b er o f d e riv a tiv e s w ith respect to its arg u m e n ts fo r a ll th e ir fin ite v a lu e s W h e n £ = the e q u a tio n ( ) is generated to (1 ) + ViX{N~ l) + + X { N) + ctNx = , x (k) = ^ T h e b e h a v io u r o f the s o lu tio n o f generated e q u a tio n e sse n tia lly depends o n the ro o ts o f the c h a c te ris tic e q u a tio n ( ) Ả N + ocX N Ằ + + Oijv- 1^ t" & N = đ ã I t is su p p o se d that the c h a c te ris tic e q u a tio n has som e p a irs o f sim p le im a g in a ry ro o ts A = 358 ± iQ a = 1,2, ,/ The rest ro o ts o f eq u a tio n (1 ) h a v e n e g a tiv e rea l p arts w ith su fficien tly g rea t v a lu es Moreover, we suppose that there is no relation o f the type (1.4) É í A = 0, s— w here q s are in tegers The o p p o s ite ca se w ill b e c o n sid er ed in the n ext p aragrap h It is easily seen th at the gen era ted e q u a tio n (1 ) h a s /-p aram eters fa m ily o f q u asip c r io d ic s o lu t io n s X = X a , COS ( o í + ự/s) , (1.5) s= where a s, \ / are arbitrary real constants ịs By virtu e o f the co n tin u o u s d e p e n d e n c e o f th e so lu tio n o n the p a m eter e, the eq u a tio n (1 ) w ith su fficien tly sm a ll £ h a s s o lu tio n s n ear to (1.5) C o n se q u en tly , w e sh all find th e so lu tio n o f eq u a tio n ( l l ) in th e series form I x = X a s COS ( Q st + ệ s) + e U j(a , (p) + e u ( a i (p) + (1 6) , 3=1 w here < = Q st + ịỊ/sì a = (flj, a 2, pa c o n ta in s i n (ps, COS (Ọt T h e y fl,)f