On the order of growth of solutions of algebraic differential equations
Assume that $f$ is an integer transcendental solution of the differential equation $P_n(z,f,f′)=P_{n−1}(z,f,f′,...,f(p))),$ $P_n, P_{n−1}$ are polynomials in all the variables, the order of $P_n$ with respect to $f$ and $f′$ is equal to $n$, and the order of $P_{n−1}$ with respect to $f, f′, ... f(p...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4584 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510733813415936 |
|---|---|
| author | Mokhonko, A. Z. Mokhonko, V. D. Мохонько, А. З. Мохонько, В. Д. Мохонько, А. З. Мохонько, В. Д. |
| author_facet | Mokhonko, A. Z. Mokhonko, V. D. Мохонько, А. З. Мохонько, В. Д. Мохонько, А. З. Мохонько, В. Д. |
| author_sort | Mokhonko, A. Z. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | Assume that $f$ is an integer transcendental solution of the differential equation $P_n(z,f,f′)=P_{n−1}(z,f,f′,...,f(p))),$ $P_n, P_{n−1}$ are polynomials in all the variables, the order of $P_n$ with respect to $f$ and $f′$ is equal to $n$, and the order of $P_{n−1}$ with respect to $f, f′, ... f(p)$ is at most $n−1$. We prove that the order $ρ$ of growth of $f$ satisfies the relation $12 ≤ ρ < ∞$. We also prove that if $ρ = 1/2$, then, for some real $η$, in the domain $\{z: η < \arg z < η+2π\} E∗$, where $E∗$ is some set of disks with the finite sum of radii, the estimate $\ln f(z) = z^{1/2}(β+o(1)),\; β ∈ C$, is true
(here, $z=\re i^{φ}, r ≥ r(φ) ≥ 0$, and if $z = \text{re } i^{φ}, r ≥ r(φ) ≥ 0$ and, on a ray $\{z: \arg z=η\}$, the relation $\ln |f(\text{re } i^{η})| = o(r^{1/2}), \; r → +∞,\; r > 0, r \bar \in \Delta$, holds, where $Δ$ is some set on the semiaxis $r > 0$ with mes $Δ < ∞$. |
| first_indexed | 2026-03-24T03:01:42Z |
| format | Article |
| fulltext |
Ys 517.925.7
A. 3. MOXOHbKO, B. ~ . MOXOHbKO (YII-T ,,J]bt~OB. nom4TexnHKa")
O IIOPYI~KE POCTA PEIIIEI-IHI~ AJIFEBPAI4-qECKHX
~[H(I)| YPABHEHHI~
Assume that f is an integer transcendental solution of the differential equation P,, ( z , f , f ' ) = Pn-t (z,
f , f " . . . . , f (P)) ,Pn, Pn-i are polynomials in all the variables, the order of Pn with respect to f and
f ' is equal to n, and the order of Pn-l with respect to f , f ' , . . . . y4p) is at most n - 1. We prove that
the order p of growth of f satisfies the relation 1/2 < p <oo. We also prove that if p = 1/2, then, for
some real 11, in the domain {z : "q < argz < 11 + 2 ~ } \ E., where E, is some set of disks with the
finite sum of radii, the estimate lnf(z) = z l /2(~ + o( 1 )), ~ ~ C, is true (here, z = r ei~, r >
> r ( t p ) > 0, andi f z = re i*, r > r(tp) > 0) and, on aray {z : argz = 11}, the relation
In [f(rei~)[ = o(r l /2 ) , r---> + ~ , r > 0 , r ~ A, holds, where A is some seton the semiaxis r > 0
with rues A < ~.
Hexatl f - - tti~Htt Tpanct~eH~tetrrmitl poan'aaOK ~HqbepeHttia, nbHOrO piaHanHa Pn ( z, f , f" ) = P, -I ( z,
f , . f . . . . , f(PJ), Pn, P~-t - - Muoroq~eHu hilt ycix aMitlmtx; cTeniub Pn Bi/~ocno f i f ' ]1opimt~ae
n, ereninb Pn-t ni/lllocno f, f ' , . . . . f(P) He nepenatUyr n - I. ~oBelleno, ttto nopmlog p apo-
c~rallna f aa/IonoJibllae nepinHocTi 1/2 < p <'0. J;lKmO p = 1/2, TO ~tJIa/leaKoro/tiiacHoro 11 a 06-
naeri {z : 11 < argz < 11 + 2 n } \ E . cnpane/t~n~ oUiHKa lnf(z) = zl/2(~l + o(1 )), z -~ **, 13
C, l~'.na z = re i~, r > r ( 9 ) > O, 12e E , ~ /leaKa Mno~mm Kpyrin i3 cKitlqelltlOIO cyMOIO
pa/liycia, a Ha npOMeHi { Z : arg Z = 11 } nHKonyeaa, ca In I f ( r e ~ ) [ = o (r I/2), r --> +**, r > 0,
r "~ A, lie A--/ teaKaMuo~Hnauaninoci r > 0 a mesa < **.
Hcno~bayeM o6oaHaqemIJ~ Teopml MepOMopdpHraX qbyHKttn~ [1]. K a r y c T a n o a n ~
Ba~tnpoa [2, c. 224] , c y ~ e c T a y ~ T a~re6panqecKHe/t~tqbqbepem~rla~bHue ypaBHenna
(n.y.) Tpea~ero nopa~Ka, aMe~atuae t temae TpanctteH~aemmue petuermJ~, nopa~OK po-
e r a ~OTOpUX p = 0. YIoa~Hee B. B. 3 H M o r z a a [3] noKaaa~, ~rro ~t.y. a T o p o r o nopa~t-
Ka He HMeeT t l e ~ x TpanctteH/IeHTHbIX pemeHn~ Hy~qeBoro nopa~Ka p o e r a . HaBeCT-
HO [4, C. 70], qrO ~a.y. nepaoro nopa/Ira
P ( z , f , f ' ) = 0 , (1)
r ~ e P ~ MHOFOqJIeH no BCeM nepeMennhlM, He ltMeeT 11e:Iux TpaHc~eH/IeHTrIUX pe-
III~HI4~I nopa/ IKa p < 1 / 2 . PaCCMOTpnM 6 o ~ e e o6u~ee no cpaBHeHmO C (1) ~.y.
( f l J ) = f j , j = 1 . . . . . p )
(I + o(1))asz%fkf: = Z bK(z)fk~ p' (2)
k+s=n IK[<n
g = . . . . . k , ) , I g l = k0 + kl + . . . + kp,
IIKII = + 2k2 + . . . + pke;
bK(Z ) = O ( z X r ) , Z ~ O = . { Z : I z l >- h } , (3)
Iz [ --> 0% k, s ~ Z , k, s > 0 , ~s ,x t r R , a s ~ C.
l ' l pe~noaoacn~ , wro Bee Koaclaclammerrna n.Y. (2) ~ a n a m r m a e c r , ne B D qbymcu~n,
Hanpm~ep, b i t ( z ) = s i n ( l / z ) , z ~ D .
MeTo~a B a v a r i a - B a m t p o n a n o a n o a ~ e T oxapaxTepHaosaTb CBOitCTBa t~e.noro pe-
tUeHnJ~ a ~ r e 6 p a a q e c K o r o ~.y. OnHUIeM acnMrrroTa~ecr, He CBOiWTBa nes~oro p e m e -
rma ~.y. (2) na npOnaBOZ~HOM ~ e {Z : arg Z = qu } H B y r ~ o a u x o 6 ~ a c T a x , qTO6U
�9 A. 3. MOXOHbKO. B. ~. MOXOHbKO, 1999
I$$N 0041-6053. Yrp. 7~am. ~'ypn.. 1999. m. 51, N ~- 1 69
70 A. 3. MOXOHIaKO, B. s MOXOHbKO
cqbopMy~tapOBaT~ COOTBeTCTBylOIIBIfl peayamTaT, rloHaRo6,qTCJt HeCKo.rlbKO onpei~e-
~IeHHII.
17ycT~ f(z) - - ttenoe pemeHme ~.y. (2) B D. O60aHaqaM m = { max s : k + s =
= n, a s ~ 0 ; k, sa Z, k , s>O} . P a a ~ e n ~ M o 6 e ~ a c T a ( 2 ) n a f n z X ' - m ' , n o c n e
npocToro npeo6paaoBamia no~yqHM
ECJqH o6oarlaqIr
(1 + o(1))(xaZ "cs-s-'~'n+m ( zfl(z) ) ~
$ = 0
= X bK(z)zm-%n (fl/f)kt ...(fp/f)kp
IKl~n-I fn-IIr
!7 s - - S - - I; m + m - - d $ ,
(4)
zf~ (z__.__) = L(z) , (5)
f ( z )
to(z) = Z bK(z) zm-x" (f l / f)kt ...(ap/
IKI ~ v- I fn-lKI
TO a.Y. (4) Momno 3alIHCaTb B Brl/Ie
m
Y~ 0 + o(X))a,z", (L(z))" = to(z),
,=0
I z l - ~ , d~=0, ~m~0.
17o neBoa qacTa (7) nocTp0HM xapaxTepacTnqecKoe ypaaHenae
m Zd s Z as LS =O.
$ = 0
~ r o ypaaHeHar HMeeT KOHeqHOe qaC~O pemenHlt
, ( 6 )
(7)
L(z) = (1.+ o(1))~jz pj, (8)
I z l - ~ , ~j=l~jlexp(iaj) , j = l , 2 . . . . . q < m ,
p j ~ paurioHaJIbHbIe qrtc~a, KOTOpbIe Haxo/IJ~TC~ C nOMOIIIblO JIoMaHHx ~jr C;
HblOTOHa [4, C. 69].
Ecam n (8) pj > 0, TO/XJIa ]xamnax pj n aj cymecTByeT KOHeHHOC MHO)KeCTBO
uemax auaqerma k, Hanpm~ep, k r { 0, 1 . . . . . mj }, TaxHx, wro ar ia a n c e ~
2gk - aj
9j = 9 ( ] , k) = (9)
Pj
BbnaoalrlalOTCZ HepaBeneTBa 0 < 9 j < 2/~. HosloT, cd~4
~ j = ~ U , k ) = ~ j - , ~ j=~U,k)= ~ j + - - , (10)
2pj 29j
j = l , 2 . . . . . q, k=O, 1 . . . . . mj.
Ypannenne (2) y a o ~ e r a o p ~ r y c ~ o s r t ~ , npH Korop~ax K ero t~eamal pememtaM
n p r ~ e r m ~ M~ro~t BH~ana - B a s m p o H a [2, c. 222]. B qac rnoe rn , n a n e e ~ o , wro mo-
6or lle~loe pemenrle f a.y. (2) runeer Ir n o p ~ t o x poeTa p = p j , r/Xe p]
I$$N 0041-6053. Yxp. ,~tam. a~'y. pu., 1999, m. 51, N ~ 1
O HOP$1~KE POCTA PEUIEHHi~ AJIFEBPAHtlECKHX ... 71
o~no ri3 '~Hcen, onpeaenerm~x B (8). I lycTb {cq} ~ MItoTKeCTBO Bcex Hy~eta tteno-
ro petIIeHHJt f n e r o npoH3BO/Iahlx f l . . . . . f r Bu6epeM npOn3BO:lbHOe G > 0 H }IJIJt
Ka~]10rO CqE { Cq} HOCTpOHM 3aMKHyTb/IR Kpyr C tteHTpOM Cq pa/~Hyca ~ q-=
= ]Cql -p-(~ qepe3 E , 0603HaqHM HHOmeCTBO r0qez , n p r l H a ~ e m a t t ~ x 3TRIM
KpyraM. Toraa cornacHo meopevte BasmpoHa [4, c. 87]
f:(z) clz 12~p+~ (11) <
z e D \ E , , c = c o n s t , j = l , 2 . . . . . p .
fI~Iz K a x a o r o Cq E { Cq} IIOCTpOHM TaKx~.e anTepBan [ ICq[ - ~q, I Cql + ~q].
I IycTb A - - MHOmeCTBO To,aeK Ha [0, oo), nprma/memamrlx ~TVlM rmTepBa.naM.
3rqi~a-uBaa (11), m e s a < ~ 25q < o o a E,--MHOmeCTBO KpyroB C KOHeaHOR
CyMHOIt pa~nycoB.
Ecnri g (z) - - t ~ e n a a (a.rm MepoMopqbHa.q) B D dpyImlma, TO O603HaqHM
M(r,g) = m a x { I g ( z ) l , z: Izl = r} . (12)
H3BeCTHO [1, C. 50], qTO ~nJt TOrO, YTO6t,I raenaz B n~tocKocTa C qbyHIO.ma g 61,ma
TpaHcI2eH/~eHTHOi~I, Heo6xo~riMO rt/~OCTaTOqHO, qT06 r4
lira l nM (r , g ) _ +o.. (13)
r~+** l n r
12enaz cl0yHKUHJt g (Z), Z e D , Ha3blBaeTc~I TpaHcueH/IeHTHOI~, ecnH BblIIOJIHSI-
eTCJ~ (13 ) .
TeopeMa 1. IIycmb f (z) ~ , e , w e mpanc,encgenmnoe peu~enue &y. (2) B D .
Tozcga f u~teem nopaOoK pocma p , 1 /2 < p < +,,~. Bbmo.anatomca ma~me c.aecgy-
1ou~ue ceoacmea:
1) c)'u~ecmeyem l > 1 yz . loe Gj = { z : "qj < argz < yj}, ~na KomopbtX 8
Gj\E,
Inf(z) = zPJ(I3jp~ I + o(1)), z --~ 0o, (14)
(o~em<apa6noztepnano argz 8 o~nacmu {z: rlj+e < argz < 7j-e}\E., ~ > 0,
~ npousoonbno sacgannoe) ;
2) cyu~ecm~ytom q> l .ay~ert {z: argz= 0j} (cmoponbt yzaoe Gj), na
KomopblX
In [f(reiO01 = o(rP), r --> o0, r E A, mes A < 0~ ; (15)
3) na cgono.~nenuu K yKaeannb~t yz:tazt u .ay~azt one ~mo~recmea E.
I f ( r e i* ) l < r v, r > r ( ~ ) , r ~ A, v =const . (16)
3/~ec~ qHcna p j n ~j , r I j n ~ IIpHHHMaIOT KaKHe-TO H3 KOHeHHblX MHOX<.eCTB
3aa~ennfl, onpe~tenerImax cooameTcamermo B (8), (10); V = max ['C K + m -
IKl~n-I
- x,~ + 2PlIKII] + e, e > 0 , e ~ K a x y r o z m o Ha.rloe.
TeopeMa 2. Ec.au &y. (2) uJteem ~ D , e . w e pemenue f ( z ) nopaSKa p =
= 1 /2 , mo cyu~ecm8yem " q j e R maroe, ~mo ~ o6nacmu { z : r l j < a r g z < r l j +
+ 2 x } \ E. eepno
I n f ( z ) = zt/2(2~3j + o(1)), z --r o., (17)
(o,en~a paono~tepna no argz e ato6oa enympennea yzaoeo~ o6aacmu), a n a ay~e
{ z : a r g z = rlj)
ISSN 0041-6053. Ytcp. ~uJm. acypn., 1999, m. M, N e I
72 A. 3. MOXOHbKO, B./I. MOXOHbKO
l n l f ( r e i r l o I = o ( r l12) , r--> +e~, r ' E ' A . (18)
3a~to tanue 1. OyaKUriJ~ f ( z ) = COS-v/-Z, Z r C, Jmn~teTca tte~naM pemenneM
a.Y. f 2 + 4 z ( f , ) 2 = 0 H HMeeT nopazoK p = 1 / 2 , nprIqeM I n f ( z ) = ~r-z(- i +
+ o(1)) , Z ~ {Z: 0 < a rgz < 2rc]., a rgz = const, T. e. cnpaaettmtBo (17), ri Ha ny-
qe {Z: a rgz = 0} manonH~e'rca (18).
3 a ~ e ~ a u u e 2. I I o K a ~ e ~ , Kax npeo6paaymTca y T a e p ~ e n r t a TeopeM 1, 2, ecml
paccMaTprmaTb 6onee mnpoKnit Knacc MepOMOpqbn~aX pemenri~. C n a q a n a npezmo-
noa:,rtM, ,-rro f ( z ) ~ MepoMopqbnoe pemenne a.Y. (1) B D . H n a x > 0 nono~rIM
ln+x = max ( l n x , 0). PaccMoTprIM rleBannrmaOaCKym xapaKreprlcTHKy m ( r , f ) =
= (2n)_ ~ ~ n in + [ f ( re i~) I d g . B [5] 6~ano noKaaaHo, qTO a paccMaTpaBaeMoM cny-
,~ae ~m6o Bepno
m ( r , f ) < l n + M ( r , f ) < v l n r , t E A , m e s A < ~ * , (19)
v > 0, v ~ HeKoTopaa KOnCTaaTa, nn6o xapaKTeprICTHKa m ( r , f ) nMeeT nopa~OK
p, 1/2 < p < ~ ; ecnH p = 1 /2 , TO BUnOZHaIOTC.a y'rBepa~]xemtJ~ (17), (18), r a e
g . ~ MHO)KeCTBO KpyFOB C KoHeqHo~ cyblMO~l paRrIycoB, HO C ReHTpaMH B ny~J~X H
n o ~ o c a x pe tuennz f .
Flopaz~OK pocTa ~epoMopqbnot4 qbyHKmIri f coBna~aeT c nopz/~OM p ee xapaK-
reprlCTrlKrl T ( r , f ) [1, c. 65]. Hop~I/~OKpOCTa xapaKrepncTnl~i4 m ( r , f ) O6oanaqrlM
qepe3 p . EcJ1rI, B qaCTHOCTri, f ( z ) ~ 12e~Ioe petuenne (1), onpeztezeHnoe B C , TO
m ( r , f ) = T ( r , f ) , p = p , a n3 (19) c~e~ayeT, qTO ~n60 f ~ ~noroqneH r ae
Bume v , m~6o f ~ Tpanctten~eHTHaZ u e n a z qbynKUnZ nopzaKa p _> 1 /2 .
Eczn f ( z ) ~ MepoMopqbnoe B yrnoBO~t o6nacTn B = { z : a -< arg z <- 13 } pe-
meHne ~t.y. (2) moHeqHoro nopz~Ka p , TO, KaK noKaaano B [6], B o6naca~a B TaK~e
BUrtO~HJ~OTC~ HeKOTOpb~e yTBep~eHr t a , aHa~IOrHqH~e cqbopMym4po~anm, tM B nyH-
KTaX 1 --3 TeOpeMu 1 gannol~ CTaTbH, 3a cne/~y~o~HM HcKn~oqem~eM: B TeopeMe 1 r~3
[6] ne yTsep~t taeTca , qTO cymecTByeT no xpat~ael~ Mepe ozma y r n o s a a o6nacT~ H
XOTJ~ 6U O/trill nyq, Ha KOTOpblX COOTBOTCTBeHHO B u n o n u z e r c a (14) n (15). 3TO OT-
nnqrie orpaz~aeT Oco6eHHOCTri MepoMopqbHb~X pemeHnt~. Hanpr Inep , pe tuenHe~
ypaaneanz ( f , ) 2 = 4 f 3 + g 2 f + g3, gj = const , ZB~ZeTCZ annrtnrnqecKa,a qbyHK-
raria BeileptuTpacca p (z) , z e C [7, c. 359], a pememIaMrt ypaBHeHri~t HeaneBe
f " = 6 f 2 + z rI f " = 2 f 3 + z f + a , a = const , ZBna~OTCa Tpaacttea/IenTHue
qbyHKttnn FleHneBe [8]. 3Tri ypaaHeHnZ nMe~OT aria (2), a HX pe tuerma aBna~OTCa
MepOMOpClDHHMH dpyHKl.r;rlZMH KOHeqHOFO rtopJt/~Ka pOCTa p [9] n n n z unx BhIIIOJIH$I-
e'rcJ~ otteaKa (c~. [6] ) :
I f (z) l < lzl C \E , , IzI > > R, v = c o n s t > 0 , (19 ' )
E . ~ aeKoTopoe r, mo;~eCTBO KpyroB C KoneqHo~ CyMMOfl pa/mycoa. OTClO/Ia c.rte-
]~yer, qXO BO Bcei~I o6nacT~ cyI/~eCTBOBaHH}t peuIeHn~I nMeCT MeCTO (16).
OcTaeTcH HeBbI$1CHeHHIalM, cymecTBylOT .rl14 MepoMopdpHne pemeHHZ ~a.y. (2),
Hlvtelolliae 6eCKOHeqHbI~ nop.'~IOK pOCTft~, HerlaBeCTHa TaK~Ke TOqHa.,q onerma CHH3y
nopaaKa p o e r a TaraX petueHnia.
~ o r a ~ m e n ~ m ~ o m e o p e ~ 1 u 2. Ha (3) , (6), (11) cneayeT, qTO cymeCTByeT
V ~ = c0nst > 0, onpe~aenJ~eMaa no Brtay ypa.aHeHUa (2), Tara.a, q-ro
,. / kp
.(fp f ) I < Izl D \E , . (20)
B~a6epeM V = const > v l a pacc~oTpm4 MHO~eCTBa
ISSN 0041-6053, Yrp. ~lam. a~'ypn., 1999, ra. 51, N ~ 1
0 I'IOP~I~IKE POCTA PEIIIEHHI~ A3"IFEBPAHqECKHX ... 73
El = {z: z~ D\E,, [f(z)l-> [z[V}, E = D\{EI[JE.}. (21)
YqHT~Ba~ (4), (5), (20), (21), no.ny~aeM (d m = O, cx m ~ O)
m
(I + o(1))Ctszd"(L(s)) a = o(I), Z~ E l , [Z[ -'~ oo. (22)
s=0
Ecyn~ ypaBaeHHe (22) He 3flBHCHT OT L, TO B J'leBOfl qaCTH n.Y. (2) TOJIbKO O~HO c~a-
raeMoe (1 + o (1 ) ) a 0 z % f n, a o ~ O, HMCCT rlo f g f l cTerlerlb n. Tor~ta cymecT-
ByeT R 1 > 0 : E l N { z : [z [ > Rl } = 0 . ~eltCTBrITe~bHO, HHaqe ypaBneuae (22)
nMeeTBna a0(1 + o(1)) = o (1 ) , z e E l , z -~ 0-, a3Haqrrr, a 0 = 0, wro npoTH-
aopeqHT npe~x_noJIoacenH~, l-loaroMy H3 (21) cze]xyeT I f ( z ) l -< I z 1", z ~ D \ E , ,
[ z [ > Rl, T.e. a paccMaTpHBaeMoM c~yqae BO Bcet~ 06~aCT~ D HMeew MeCTO yTaep-
~ e H H e 3 reopeMta 1. (OTMeTHM, qZO B npHHepax, paccHoTpeHHUX B 3aMeqamm 2,
xapaKTepHCTHqeCKHe ypaBHenHJ~ He 3aBHCJ~T OT L, HO3TOHy ~J~ HHX rt BblrlO.rlH~qeT-
CA OUeHKa (19 ' ) . )
Ec.rm ypaBHeHrie (22) 3aBHCHT OT L, L = z f l ( z ) / f ( z ) , TO OHO rlMeeT KOHeqHoe
qHCnO pemeHHfl [4., c. 69]
zA (z) = (~ j+o(1 ) )Z pj, z ~ E I , j = l , 2 . . . . . q, [z[---> ~ . (23)
f ( z )
l-lycT~ Z. ~ E l , [Z, [ > r 0 > h. PaCCMOTpHM CBZaHy~O KOMnOHeHTy E 0, Z.
E 0 c E 1 . B ~a~r~o~ TOaKe Z r E 0, I z [ > r0, r0 ~/XOCTaTO~HO 6 o ~ , m o e , Bta-
no~HaeTca O~HO H3 COOTHOtUeHH~ (23). HH B OZ~HOfi rq~Ke Z ~ E 0 He ~ o r y r 6~aT~,
paBH/.dMH npaB~e ~aCTH z~Byx cOOTHOmeBHfl Ha (23) C HHReKCaMH ,* H j TaKHMr~, qTO
l i t - [~j[+ [ P t - Pj[ > 0. B ToqKe Z. ~ E o c E I Blano.rlHaeTca ( 2 3 ) c HeKOTOpt, I-
MH qbHKcHpOBam~Ma j . Y~HTtaBaZ aenpepUBHOCTb Z i f ( Z ) / f ( z ) , nozy~aeM, qTO
TO ace COOTaomeHHe BbIrlOJIH~qeTCJl Z~:~l Ka~/~OrO Z ~ E0, ] Z [ > r0. I-IO~TOMy 143
(23) cne/xyeT, wro npH Ka~r e > 0 cytuec'myeT r o > 0 Tahoe, aTO
z f ' ( z ) = (~ + u(z))z p, z ~ Eo, lu(z)l < a, (24)
f ( z )
Izl > re, u(z ) ~HeKOTOpa~l qbyHKl.IH~, ~ = I3(E0), p = P (E0) ; 13, p ~ o z m o H3
qHce.rl ~j, pj. 0603naqHM
M ( r ) = M ( r , f ) = max If(z)l = l / ( ~ ) l , ~ = r e x p ( i c p ( r ) ) . (25)
Izl--r
HO qbop~yne MaKHHTa~pa [4, C. 62] B TOq~Ce MaICcH~y~a ~, [ ~ [ = r,
;f'(;_.__~) = rM'(r.___~) = K( r ) , (26)
f (~) M (r)
M ' ( r ) ~ IIpOHZBO]~Ha51 cnpa~a. I'Iocxo,rlbKy ~ I lle~olt rpaHct~ertaerrrao~ cIDyI'IKI/,HH
In M ( r ) / I n r---> **, K ( r ) --> + **, r ---> ** [4, c. 66], TO H3 (21) cJ-Ie~yeT, q t O ;
r E 1 , [ ~ [ = r > r ' , a yq~T~Ba~ (24), (26), noay~ae~
K ( r ) = (1 + o(1))[~s~ p" > O, Ps > o, I~1 = r ~ A, (27)
m e s a <**, s = l , 2 . . . . . q ( cM. ( l l ) ) . rIyc-n,
ISSN 0041-6053. Yxp. s~m. ~'vpa., 1999, m. 51, I~- I
74 A. 3. MOXOHbKO, B. ~. MOXOHbKO
p . = lim ln:-'r'/((~ _< ~ ( ~ l i m l n K ' r " = p*, r - e ~ o .
l n r ln r
In K ( r )
Ha (27) cneRyeT, wro BHe MHO~KeCTBa A XOHeqHO~ Mepr40TnOtUeHne ~ npri
l n r
r --~ co rIMeeT KOneqnoe qncno npe/IeamHHX 3Ha~emti~ p i . . . . , pq. OTC~O/Ia B~aTexa-
eT [4, C. 33], neMMa 2,23 n cne~cTBae 1, '~TO p . -" p* = p , p coBna,aaeT c o a n n ~
aa Ps B (27). TaKriM o6paao~, cytraecTByeT lim InK( r ) = ln r p~ , r---> o . , r ~ A ,
mes A < o.. 1-10aTO~y rI3 (27) cne~yeT ([4, c. 34], ner4Ma 2.33)
K ( r ) = (11 ,1 + o(1)) rp, r > r ' ,
P, [~s [ OaHO n TO ~ e a n n Kazxaoro r > r ' . E c n n O603HaqaTb [~s
= [ [31 exp (its), ~ = r exp (i 9 ( r ) ) , To na (27), (28) nozyqaeM
pq~(r) + a = 2r~k + o ( 1 ) , cos (pq~( r ) + o~) = 1 + o ( 1 ) ,
1
c o s ( p g ( r ) + a ) > ~ , r E A , m e s A < ~ , r > r ' ,
(28)
= ~ =
(29)
k ~ Ltenoe, p = p s - const , '9' r > r ' ; a , k npnHrIMatOT KoHeqHoe qrtca/o
r M ' ( r )
BO3MO)KI-.IblX 3HaqeHnlt (0 < ~ ( r ) < 2~x). I,'I3 (28) rt COOTHOmenI4g K ( r ) =
M ( r )
nwrerpnpoBaHHeM no,ny~aeM
l n .M( r ) = ( l ~ lP - I + o(1)) P , r > r ' , p > 0 , (30)
p n IIB I n To JKe nprl gaa~oM r > r ' . Ha (30), (21) c~e/IyeT
I f ( ; ) l = M ( r ) > r v, ~ E 0 = E o ( r ) c E I , (31)
E 0 ~ CB~3Ha~I KOMHOHeHTa E l . O603HaHHM
2 ~ -- CZ n 2~k -_._____~ +
1"1 = p 2 p ' y : P ~p , (32)
r~e k = k ( r ) ~ ttenoe qrtcno, onpe~eneHHoe n (29).
HOKaT~eM, qTO
VT~, ~ ( r 1 < 7 ~ < ~ < 7 ) ,
{z: Izl = r , :Z -< argz -< V } c E o, r > r ( Z , V ) . (33)
Ecnrt rl < Z -< 0, < ~ < 7, To, yqrrr~maz (32), (30) (p > 0), naxo~HM
T i p + a = 2 r ~ k - E < p x + a _ <
2
7[
< p C , + a < p ~ / + ~ < TP + a = 2r,..k + ~..
IIOaTO~4y
cos (p 0, + a ) -> rain (cos (p ~ + a ) , cos (p V + a ) ) > 0. (34)
H y c r o 00 - - HalIMCHbmCC, e . - - Han6oJIbmec 3HaqCHH~I TaKHC, HTO ~yra h =
= { z : z = r c x p ( i O ) , 0 0 ~ 0 < 0 , } c E 0. H3 (31), (21) cnc~ycT, qTO 0 0 <
< arg~ < e , . HnTcrp~rpya (24) no/~3're h OTTOqKn ~ aOTOqKn Z = r c x p ( i ~ 0 ) ~
ISSN 0041-6053. Y~p. ~tam. ~.'vtm.,1999 , m. 51, N ~ 1
O rlOP.q~KE POCTA PEUIEHHI~ AJII'EBPAHHECKHX ... 75
e h, B ~ e n a a ]~el~C'l'BHTeJIbHIde qacrlt H yqHThIBas, HT0 COS (p (argO) + oc ) =
= 1 + o(1) (CM. (29)),nonyqacM
In f(reiCO)l [ [ ~ [ p - l r P [ c o s ( p i D + a ) - c o s ( p ( a r g ~ ) + a ) + o (1 ) ] =
f ( ~ ) l
= [ [ 3 [ p - i r P [ c o s ( p i D + a ) - 1 + o (1 ) ] .
IIocxoJmKy g ( r ) = If(~) l, TO OTC~na H H3 (30) CaenyeT
ln[ f ( re i~)[ = [ ~ l p - l r P [ c o s ( p i D + a ) - 1 + o(1) ] +
+ I~l P - l r P ( 1 + o (1) ) =
= I[~]p-lrP[cos(piD + a) + o(1)], 00<iD<0. . (35)
YIpczmoaoacuM, qT0 0, ~ W" 143 OI~IpC~cJ"IOHH~ CB.,~aHO~ KOMI'IOHCHT]~ E 0 H onpc-
~CaCHHS 0 . cnc~yc~r (cM. (21)) [f(reiO*)[ = r v. HOaTO~ty, ccnH n (35) q) = 0 . , TO
n o n y , a ~
l n r v = [ B l p - l r P [ c o s ( p 0 . + ~) + o(1)].
Y,~trn~as (34), B ~ M , wro ~TO paBCHCTBO npPl ~ocraTo'mo 6oJ~stuHx r HCBOaMO)K-
HO. I'Io~ToMy 0 . > V" AHanorHqHO MO~O HOKaSaTb, wro 00< Z. TeM caMH~ (33)
~oKaaaHo. Tor~a via (35) cnc~yeT
lnlf(reiCP)[ = l [ 3 [ p - ' r P [ c o s ( p i D + c~) + o (1 ) ] , Z<-ID<-~/. (36)
Hpe~nonoaCZ~M, qToa(36) p < l / 2 . Tor~a y - v l = Jz/p > 2~ H qHcna Z H V
MO~L'~O auSpaTb TaK, WrO6,., Blm]OnHanHCb yCnOBHa
s i n ( p z + a + ~ p ) ~ 0, W = Z + 2 ~ . (37)
[IOCKOnbKy f ( r e iz) = f(rei(Z+2x)), TO Ha (36), (37) caenycT
0 = In [ f ( re iu - in [ f (reiZ)[ =
= ( c o s ( p z + 2~p + a ) - c o s ( p z + a ) + o(1)) l~ l rPp -~ =
= ( - 2 ( s i n ~ p ) s i n ( p z + a + x p ) + o(1)) l [~ lp-~r p. (38)
IIocK0a,.zy 0 < X p < = / 2 , TO sin ~z p ~ 0. Tor~a, y .HT~Saa (37), nony . ac~
s in~p sin(p Z + a + ~ p ) ~ 0, qTO npoT~nope~mv (38). TaZHMO6paaOM, p > 1/2.
CymecTnyeT KOHCqH0e '-IHCnO aHaqCHHfl ida e [ 0, 2~], RJIa KOT0pI~IX
cos(piDs + a)= 1, aH~em~o (ps =(2~k- cc)/p, (k, p, a npHHm~amT KOHe'~-
HOe HHCJIO 3HaqeHHJ~). COOTBeTCrBCHHO cy~eMrBycT KOHeqHOe W/4CJIO HHTepBan0B
[iDs-=/3p,iD, +~/3p] = [Xs,~ts], HaKOTOpHX
1 ~,~ < 0 < lXa, 'S e I, (39) cos(p e + ~) > ~,
rae I -zoHeqHOe smoxe~o m~ezco~.
Ha (29) cae~,~r, ,fro ~ ~ = r exp ( i qU (r)) -- Toga .azcm~yua, TO a~aqcm~e
iD(r) npm~a~e~rr Zaz0My-ro Ha ozp~os [~.~, tt~]:
( V r > r ' , r ~ A ) ( ~ s = s ( r ) e I ) : ~., < iD(r) < l~ ,
[ f ( r e x p (iq~(r)) [ ffi M(r ) . (40)
ISSN 0041.6053. Y~p. :,am..:@,p... 1999. m. 51. !~ 1
76 A. 3. MOXOHbKO, B. ,/1. MOXOHbKO
rIOCKOJIbKy OTpC3KOB [~'s, Its], s e I, KOItCqrlOe q14CYI0, TO
( 3 s ( O ) e l ) ( V a > 0 ) ( 3 r > a , r ~ A ) : ~,s(0) < (p(r) < Its(o). (41)
IIycT~ A = { r : r ~ A , Xs(O) < (p(r) < Its(o)}. H3(41) CnvATeT supA = +0o.
Tor~a Ha oTpC3KC [~'s(0), Its(0)] C ] X, V[ BblnOJIH~CTC~ cOOTHOmOHrIC (36) :
In ]f(reie) l - I [3[ p-lrP cos (p 0 + a),
(42)
~s(o) < 0 < ~ts(o), r e A.
B [6, c. 518] 6bino/~ozazar[o TaKOe yrsep~,~enae: ecJm aJm MHo>zec'rBa KpyroB E ,
c ReHT'paMH B B TOqKaX Cq, Cq fi { Cq}, H pazmycaMa 8 q Bblrlo.r/H~eTC..q Z ~q < C
(CM.(l l)) ,TO V 0 t , 0 2 , 0 < 0 t < 0 z < 2 r c , 3~:, O t < K < 0 2, TaKOe, aTO .nya {Z:
argz = ~:, I zl > a > o} He nepeceKaeT Kpyra E , : {z : argz = ~:, I zl > a =
= 2 z c / ( O 2 - 0 1 ) } I ~ E . = O.
HoaTo~y, ec~tHBu6paTb 0 t = ~'s(0), 02 = Its(0), TO 3 K , ks(0) < ~ < Its(0):
zy,~ s(~:) = {z : argz = ~ , Izl > d = 2rcc / ( i t s (o) - Xs(0))} nMeeT CBO~CTBO
s ( ~ ) 13 E, = o .
H3 (41), (42), (39), (21) c.rte/xyeT, '-fro E l A S('~) :~ O . MHO~ec'rBo E 1 I'q S(K)
MO}KHO Hpe/IcTaBHTb m Bri/Ie 06"bO/XHHOHHJI ,,MaKCHM831bHIalX" oTpe3KO B COt TBKHX,
qTO
I f (z ) l -> I zl ~, z e co,, (43)
rlpHqCM, ec.rlH Zl t - -Haqa.no, Z2t ~KOHeR cot I4 ]Zlt j > r', IZzt] < 0% TO
[ f (z t , ) [ = [zt,[ v, If(z2,)[ = [Z2, l v. (44)
PaCCMOTpHM Ha ~ e S ( ~ ) TOqZy Z = r , e iz, r , e A. H3 (39), (42), (43) c.ne-
/1yeT, trro r, ei~:e cot C E o , EO ~CBZaHaa KoMnoHeHTa E t . Ha cot B~nO.nH~aercz
(24). EcJm IlpOrlHTerpnpOBaTt, (24) Ha cot a Bra,/~eJlH-rl, ]~et~c'rBnTe.rlbHHe ttaCTH, IIO-
~y'a~ard ( q ( z ) , v (Z) ~ HeZOTOpUe dl)yrmtm14, Iq(z) l, l u (z ) l < ~, ~5 > 0 - - gag
yro~-~o ~4anoe, ec.na I z I aocraTO'aHO 6OnbLUOe, Z~r = Z~, Z2t = Z2, } Zi I = r t ,
Izz l = "z) , To
hi [ (z) = (z p _zlp)(~p-1 + 1)(Z)), Z = r e i ~ e cot, (45)
f(Zl)
I : (0 [ = ( : _ ~)(II~I p-' cosCpz + a) + q(0), r~ -< r _< r2. (46)
IIpmm~a.,t so BHmdaHHe (39), (42), no.ny'.lac~ cos (p Z: + a) > 1/2. I'IpcAno~IO~M,
q'ro cot m4eeT KOHe,myIO Aa14Iiy. Ha (44), (46) cJzcAyeT V In.(r2/r I ) = ( ~ -- rl p) X
x (ItBlp-~cosCo~: + a) + e(z~)), nOaTO~y vp-~In (r~/r~) > (r~ - ~)ll~IP -~ x
x 2-~cos(p~ + a) 1 4 ~
2v . (47)
c ( l n x 2 - l n x 0 > x= - xt , x~ = ~ < x~ = ~ , c = It3lcos(p~: + c0
O y m o ~ a x - c l n x aoapacTaeT Ha It ,**), IIOaTOMy HepaBeHCTBO (47) HeBOZMO-
~ 0 , r r I (C,rlCROBaT~/IBHO, Xl) /~oc'raToqllo 6OJIblIIO~. OTclo]la cJIe]lyffr, qTO
155N 0041-6053. Yrp. ~am. :~. pn . , 1999 , m. $1,1V ~ ]
O FIOPJ;UIKE POC~A PEIIIEHHI~I AJIFEBPAHqECKHX ... 77
l n f ( z ) =
In If(z) l =
TaKa~ o6pa3oM,
(J0 t HMeCT 6eCKOHCqHylO RJIHHy, CCJIH t i t ,/IOCTaTOqHO 6OJll, IIIOC.
(45), noJ~yqaeM
Z p ~ p - ! ( 1 + o ( 1 ) ) , Z = re i~, r > r O,
rP[l~[ p-lcos(p~: + a)(1 + o(1)), r > r o.
ToF~a , yHHTHBa~
(48)
{ z : argz = ~:, Iz[ > r0} c E 0. (49)
IIyc-n, B (32) r p, g TO 3aaqenH,q, npH KOTOpHX B (29), (39), (41) onpe/Ie.rlaeTc.q
oTpeaoK [~s(0), ~ts(0)]- IlosTopaa c aeanaqaTenbau~m naMeaenaaMH p a c c y ~ e -
HH~q, HCnO.rroaoaaHHUe n p n AOKaaaTe2IbCTBe (36) (aTO RoKaaaTe2II, CTBO CM. B [6,
qbOpM. ( 4 4 ) ] ) , MO~KHO noKaaaTb, qTO Ha (49) Cnc~ycT: V Z , V (13 < Z < ~g < T),
{ z : Z -< a rg z -< V , [ z I > r0 } \ E , c E 0 n a aTOR o 6 n a c T a Buno~maCTCa o a e n g a
(14), paaHoMcpHaa no arg z. Ec.aH s (32), (14 ) ) ' - 13 = 2~, TO Ha (14) c~caye'r (17).
,~OKa3aTCXIbCTBO OI.~CHOK (15), (16) CMOTpH a [6].
1. Fo,~6epz A. A., Ocrapoocrul~ I4. B. PacnpeAcneHue anaqeHa~t MepoMopqbtl~x qbylmuHtl. - M.:
HayKa, 1970. - 5 9 2 c.
2. Ba.~upon )K. AllaJml~qecKne qbylmttHn. -- M.: FocTeXTeopna/laT, 1957. - 235 c.
3. 3u,~wen,~i~ B. B. 0 nopa/IKe pocTa RCdlblX TpalICRelIAeHTIlUX pemellrll~ a/Ire6parlqeCKnX /Iaqbqbe-
pellanaJIbH~X ypaunelmlt nToporo nopa•Ka // MaT. c6. - 1971. - 85(127), N~2 (6). - C. 286 - 302.
4. Cmpe,~uq 111. H. ACHMrlTOTHqr CBOgCTBa atlaJ1HTHqeCKHX pemenH• ~H~pClIRHaJlbHHX
ypaBHeHHIL- BrtnblUOC: Mmrrnc, 1972. - 4 6 7 c.
5. Moxon~o B. ~[.. Fopfar A. E. 0 MepoMopqbitux pemeHa~X ~mqx~pemmam,nux ypaane-
tatar1 nepaoro nopa/tKa///IHqbqbepeHu, ypalmeHHa.- 1991. - 2 7 , N e 6. - C . 1087-1089.
6. MoxoabKO A. 3. 0 MepoMopqbHhlX pemeHn.qx a.nre6paaqecgHx ~[a~pCHuHa3"lbHbig ypaanenafl n
yr.~oaux o6oaacTax//Ygp. MaT. a~ypn. - 1992.- 44, N -~ 4 . - C. 514-523.
7. Map~.'vtueeuq A. 14. TcopHa aHa.aaraqccxnx qbynguatL B 2-x T. -- M.: Hayga, 1968. - T. 2. -
624 c.
8. Fo,~y6eo B. 8. Jlegunn no aHa.amlaqecxo~ Teopaa Anqbqbepemtaa.nbnux ypaaaenalt. - M.; JI.:
rocTexTeopaa~laT, 1950. - 436 c.
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noc..ae Aopa6OTKH - - 12.03.98
ISSN 0041-6053. Yrp. ~tam. ,,rypx., 1999, m. 51, bl ~ 1
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| id | umjimathkievua-article-4584 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:01:42Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f2/0f5510e671426e527bf602b9799d46f2.pdf |
| spelling | umjimathkievua-article-45842020-03-18T21:09:14Z On the order of growth of solutions of algebraic differential equations О порядке роста решений алгебраических дифференциальных уравнений Mokhonko, A. Z. Mokhonko, V. D. Мохонько, А. З. Мохонько, В. Д. Мохонько, А. З. Мохонько, В. Д. Assume that $f$ is an integer transcendental solution of the differential equation $P_n(z,f,f′)=P_{n−1}(z,f,f′,...,f(p))),$ $P_n, P_{n−1}$ are polynomials in all the variables, the order of $P_n$ with respect to $f$ and $f′$ is equal to $n$, and the order of $P_{n−1}$ with respect to $f, f′, ... f(p)$ is at most $n−1$. We prove that the order $ρ$ of growth of $f$ satisfies the relation $12 ≤ ρ < ∞$. We also prove that if $ρ = 1/2$, then, for some real $η$, in the domain $\{z: η < \arg z < η+2π\} E∗$, where $E∗$ is some set of disks with the finite sum of radii, the estimate $\ln f(z) = z^{1/2}(β+o(1)),\; β ∈ C$, is true (here, $z=\re i^{φ}, r ≥ r(φ) ≥ 0$, and if $z = \text{re } i^{φ}, r ≥ r(φ) ≥ 0$ and, on a ray $\{z: \arg z=η\}$, the relation $\ln |f(\text{re } i^{η})| = o(r^{1/2}), \; r → +∞,\; r > 0, r \bar \in \Delta$, holds, where $Δ$ is some set on the semiaxis $r > 0$ with mes $Δ < ∞$. Нехай $f$—цілий трансцендентний розв'язок диференціального рівняння $P_n(z,f,f′)=P_{n−1}(z,f,f′,...,f(p))),$ $P_n, P_{n−1}$—многочлени від усіх змінних; степінь $P_n$ відносно $f$ і $f′$ дорівнює $n$, степінь $P_{n−1}$ відносно $f, f′, ... f(p)$ не перевищує $n−1$. Доведено,що порядок $ρ$ зростання $f$ задовольняє нерівності $12 ≤ ρ < ∞$. Якщо $ρ = 1/2$, то для деякого дійсного $η$ в області $\{z: η < \arg z < η+2π\} E∗$, справедлива оцінка $\ln f(z) = z^{1/2}(β+o(1)),\; β ∈ C$, для $z=\text{re } i^{φ}, r ≥ r(φ) ≥ 0$, де $E∗$ — деяка множина кругів із скінченною сумою радіусів, а на промені $\{z: \arg z=η\}$ виконується $\ln |f(\text{re } i^{η})| = o(r^{1/2}), \; r → +∞,\; r > 0, r \bar \in \Delta$, де $Δ$—деяка множина на півосі $r > 0$ з mes $Δ < ∞$. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4584 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 69–77 Український математичний журнал; Том 51 № 1 (1999); 69–77 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4584/5872 https://umj.imath.kiev.ua/index.php/umj/article/view/4584/5873 Copyright (c) 1999 Mokhonko A. Z.; Mokhonko V. D. |
| spellingShingle | Mokhonko, A. Z. Mokhonko, V. D. Мохонько, А. З. Мохонько, В. Д. Мохонько, А. З. Мохонько, В. Д. On the order of growth of solutions of algebraic differential equations |
| title | On the order of growth of solutions of algebraic differential equations |
| title_alt | О порядке роста решений алгебраических дифференциальных уравнений |
| title_full | On the order of growth of solutions of algebraic differential equations |
| title_fullStr | On the order of growth of solutions of algebraic differential equations |
| title_full_unstemmed | On the order of growth of solutions of algebraic differential equations |
| title_short | On the order of growth of solutions of algebraic differential equations |
| title_sort | on the order of growth of solutions of algebraic differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4584 |
| work_keys_str_mv | AT mokhonkoaz ontheorderofgrowthofsolutionsofalgebraicdifferentialequations AT mokhonkovd ontheorderofgrowthofsolutionsofalgebraicdifferentialequations AT mohonʹkoaz ontheorderofgrowthofsolutionsofalgebraicdifferentialequations AT mohonʹkovd ontheorderofgrowthofsolutionsofalgebraicdifferentialequations AT mohonʹkoaz ontheorderofgrowthofsolutionsofalgebraicdifferentialequations AT mohonʹkovd ontheorderofgrowthofsolutionsofalgebraicdifferentialequations AT mokhonkoaz oporâdkerostarešenijalgebraičeskihdifferencialʹnyhuravnenij AT mokhonkovd oporâdkerostarešenijalgebraičeskihdifferencialʹnyhuravnenij AT mohonʹkoaz oporâdkerostarešenijalgebraičeskihdifferencialʹnyhuravnenij AT mohonʹkovd oporâdkerostarešenijalgebraičeskihdifferencialʹnyhuravnenij AT mohonʹkoaz oporâdkerostarešenijalgebraičeskihdifferencialʹnyhuravnenij AT mohonʹkovd oporâdkerostarešenijalgebraičeskihdifferencialʹnyhuravnenij |