Solution of a Class of Riccati Equations

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1 Proceedings of he 8h WSEAS Inernionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp ) Soluion of Clss of Ricci Equions K BUSAWON, P JOHNSON School of Compuing, Engineering nd Informion Sciences Norhumbri Universiy Ellison Building, Newcsle-upon-Tyne, NE 8ST UNITED KINGDOM krishnbuswon@unncuk Absrc: In his pper, we derive he nlyicl closed form soluion of clss of Ricci equions Since i is well known h o every Ricci equion here corresponds liner homogeneous ordinry differenil equion (ODE), he resul obined is subsequenly employed o derive he nlyicl soluion of he clss of second order liner homogeneous ODEs corresponding o he clss of Ricci equions considered In ddiion, we presen some exmples in order o demonsre he vlidiy of he resuls obined Inroducion A lrge clss of dynmicl sysems ppering hroughou he field of pplied mhemics re described by ODEs of order of he following form: ẍ()+p()ẋ()+q()x() =r() () where ẋ = dx d, ẍ = d x,ndp(), q() nd r() re rel d vlued, sclr funcions of he independen vrible I is no obvious o perform complee nlysis of he equion () since here is no universl mehod for solving he liner inhomogeneous ODE of order There do however exis well esblished mehods for solving () in number of specil cses, one of which being when p() nd q() re consn [] A survey of he vrious exising mehods (nlyicl nd oherwise) for solving he ODE () is given in mny reference exbooks (see eg []-[0]) On he oher hnd, i is well known h solving he ODE () in he homogeneous cse when r() =0, mouns o solving Ricci equion (see eg []) More precisely, he following second order homogeneous ODE ẍ()+p()ẋ()+q()x() =0 () cn be rnsformed ino he following Ricci equion ẏ()+p()y() y () =q() (3) by mking he following chnge of vrible: y() = ẋ() x() (4) In generl, i is no ll simple mer o solve (3) for rbirry funcions p() nd q() This implies h he rnsformion (4) does no mke i ny esier o find he soluion of () However, s Euler ws ble o demonsre, when one priculr soluion of (3) is obined, he generl soluion cn be redily found [] The difficuly in using (3) o obin he soluion of () herefore resides in he problem of finding priculr soluion of (3) I ough o be sressed h when p() nd q() ssume some specil form, hen he generl soluion of (3) cn be found For exmple, when q() 0, he bove Ricci equion reduces o he Bernoulli differenil equion nd hus cn be reduced o firs order liner ODE by mens of he rnsformion y() = which yields v() v() =p()v() nd whose soluion is given by v() =v( )e e λ where is n iniil vlue of In oher words, when q() 0 he soluion of (3) cn be explicily wrien s y( )e y() = y( ) λ (5) In his pper, we re specil cse of he Ricci equion (3) where he funcion q() is no ideniclly zero for ll More precisely, we consider he cse where he funcion q() is posiive for ll nd is given by q () = e +K e λ dλ

2 Proceedings of he 8h WSEAS Inernionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp ) where = q ( ) > 0 nd K is consn We shll give he explici soluion of he Ricci equion (3) when q() is s bove Since q () depends on p(), we shll ssume, hroughou he pper, h he inegrl exiss We shll lso use he resul obined o provide he soluion of he liner second order ODE corresponding o he Ricci equion considered As fr s he uhors re wre, he explici soluion of he clss of ODEs considered here does no exis in he lierure This lys he foundion for he nlysis for h priculr clss of ODEs in erms of sbiliy, exisence of singulriies, exisence of periodic soluions nd so on An ouline of he pper is s follows: In he nex secion, he soluion of clss of Ricci equions is given long wih n exmple This resul is hen used o derive he soluion of he clss of second order liner homogeneous ODEs corresponding o (3) nd furher wo exmples re provided o illusre he mehod Finlly, some conclusions re drwn Soluion of Clss of Ricci Equions In his secion, we give he generl soluion of clss of Ricci equions The resul is summrised in he following heorem: Theorem Consider he Ricci equion ẏ()+p()y() y () =q() (6) wih he iniil condiion y( )=y 0 for some iniil vlue Assume h q( )= > 0 nd h he inegrl exiss Assume furher h he funcion q() sisfies he relion q () = e +K λ for some consn K Then, he generl soluion of (6) is given by y() =f() q() (7) where he funcion f() is given by K + α +e αθ () e if αθ () K > 4 f () = K + β n βθ () if K < 4 K if K =4 θ 3 () (8) nd he funcions θ n ();n =,, 3 re given by θ n () =c n + q (τ)dτ (9) where c = α ln y0 + (K α) y 0 + (0) (K + α) c = y0 + K β n β () = y 0 + K () nd α = K 4 β = 4 K Before giving he proof of he bove heorem, we shll mke few remrks on he funcions q() nd f() Remrks Firs of ll, noe h if =0hen q() =0for ll, nd, s menioned in he inroducion, in his cse he soluion is given by (5) Nex, i cn be shown h q (), whichisgiven by q0 e q () = +K λ, is he soluion of he following Bernoulli equion in q() : q() = p() q() Kq () = p() q() K q () (3) wih iniil condiion q ( )= 3 By differeniing he funcion (8), i cn be shown fer some lenghy clculions h f () = f ()+Kf ()+ q() (4) In oher words, (8) is he soluion of he bove differenil equion wih some specific iniil condiions Proof - Theorem : Le us firs demonsre h y p () =f () q () (5)

3 Proceedings of he 8h WSEAS Inernionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp ) is, les, priculr soluion of (6) By differeniing (5) we hve ẏ p () = f () q()+f () q () (6) Nex, by considering he bove remrks nd by subsiuing (4) nd (3) ino (6) we obin ẏ p () = f () q()+q() p()f () q() or lernively = y p ()+q () p () y p () ẏ p ()+p () y p () y p () =q () Hence, i is cler h (5) is indeed priculr soluion of (6) Now, he generl soluion of (6) is given by where u() = v() y() =y p ()+u() nd v() sisfies v()+(y p () p()) v() = More precisely, (y p(τ) p(τ))dτ u 0 e 0 u() = λ (y p(τ) p(τ))dτ u 0 where u 0 = u( )Ifc, c nd re ll chosen s in (0-), i cn be shown h y p ( )=y 0 Asresul, u 0 =0 I is cler from he bove h if u 0 =0,hen u() 0 Consequenly, if c, c nd in (8) re ll chosen s in (0-), hen y p () becomes he generl soluion of (6) ie y () =y p () QED Remrk Noe h for K<0, hefuncionq() nd hence he soluion x() migh possess some singulriies This will be illusred by n exmple in he nex secion Exmple Consider he iniil vlue problem ẏ 3 + y y = (7) where y ( )=y 0, =0nd where is some consn Now, he Ricci equion (7) is of he form (6) wih q () = nd p () = 3 + I is cler h q ( ) > 0 In ddiion, i cn be shown h q () sisfies he relion (3) wih K =3 From he resul of Theorem we cn sy h he generl soluion of (7) is y () = 3 5 +e 5( ( 0)+c ) + e 5( ( 0)+c ) where c = 5 ln y 0+(3 5) y 0+(3+ 5) 3 SoluionofClssofndOrder Liner Homogeneous Sysems As menioned in he inroducion, o every Ricci equion here corresponds liner nd order homogeneous equion In priculr, o he Ricci equion (3) here corresponds he homogeneous liner ODE () We cn herefore use he soluion of he previous clss of Ricci equions o derive he soluion of is corresponding nd order ODE This is given in he following corollry: Corollry Consider he iniil vlue problem ẍ()+p()ẋ()+q()x() =0 (8) wih given iniil condiions x( )= nd ẋ( )= ẋ 0 Assume h q( )= > 0 nd h he inegrl exiss Assume furher h he funcion q() sisfies he relion q () = e +K λ for some consn K Then he generl soluion o (8) is given by: e αθ () e αc e ( K α )(θ () c ) if K > 4 βθ x() = x () 0 e K (θ() c) if K < 4 βc θ3 () e K (θ 3() ) if K =4 where c, c nd re given by (0-) in which y 0 = ẋ0 Proof - Corollry : Employing he rnsform (4) in (8) yields (6) Now, by Theorem we know he soluion of (6) is given by (7) since we re ssuming h (3) holds Solving (4) for x i is cler h x () = e y(τ)dτ = x0 e f(τ) q(τ)dτ (9)

4 Proceedings of he 8h WSEAS Inernionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp ) nd hence he soluion of (8) is (9) where f() is given by (8) nd where y 0 = ẋ0 To deermine he explici form of he soluion of (8) i is necessry o deermine he following inegrl I () = f (τ) q(τ)dτ To do his we se u = q(τ)dτ Then, funcions θ n ();n =,, 3 describedin(9)cnbewrien θ n () =c n + u The funcion f () cn herefore be wrien s funcion of u s follows: K + α +e α(c +u) e if α(c +u) K > 4 f (u) = K + β β n (c + u) if K < 4 K if K =4 + u As resul, I() = f (u) du nd from his simplified form i is possible o show, fer some lenghy clculions, h α K (θ () c )+ln I () = K (θ () c )+ln e αc e αθ () βc βθ () if K > 4 if K < 4 K (θ 3() )+ln θ 3 () if K =4 This implies h e αθ () e αc e ( K α )(θ () c ) if K > 4 βθ x() = x () 0 e K (θ() c) if K < 4 QED βc θ3 () e K (θ 3() ) if K =4 Exmple Consider he iniil vlue problem ẍ + + ẋ + x =0 (0) where x ( ) =, ẋ ( ) = ẋ 0 wih > 0 nd is some consn By inspecion i cn be seen h, (0) is of he form (8) wih q () = nd p () = + I is rivil o show h q () sisfies he relion (3) wih K = From he resul of Corollry we cn sy h he generl soluion of (0) is x () = 3 ln 3c +c 0 where c = 3 n 0ẋ0+x0 3x0 Noe h he soluion possess singulriy = 0 Thebove soluionholdsslongs>0 becuse of he ln erm ppering in he soluion Exmple 3 Consider he iniil vlue problem ẍ b +e b ẋ + e b x =0 () where x ( )=, ẋ ( )=ẋ 0 for some iniil vlue nd nd b re consns The form of () is equivlen o h of (8) wih q () = e b nd p () = b +e b Agin, i is simple mer o show h q () sisfies he relion (3) wih K = Corollry shows h he soluion of () is b e b e b + c3 x () = e b (e b e b ) where = e b ẋ 0 Ke b 4 Conclusion In his pper, we hve derived he nlyicl soluion of clss of liner homogeneous ODEs of order We hve firs given he soluion of priculr clss of Ricci equions which is hen used o chrcerise he soluion of firly lrge clss of homogeneous second order liner ODEs Exmples were given in ech cse o demonsre he mehods hnd The resuls obined will provide beer insigh on he clss of dynmicl sysems considered in erms of sbiliy nd phse plne nlysis References [] R Bronson, Schum s Ouline of Theory nd Problems of Differenil Equions nd Ediion, McGrw-Hill Inc, 994 [] D Zwillinger, Hndbook of Differenil Equions, Acdemic Press Inc, 989 [3] DL Kreider e l, An Inroducion o Liner Anlysis, Addison-Wesley Inc, 966

5 Proceedings of he 8h WSEAS Inernionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp ) [4]ADPolyninndVFZisev,Hndbook of Exc Soluions for Ordinry Differenil Equions nd Ediion, Boc Ron, Chpmn & Hll/CRC Press, 003 [5] AC King l, Differenil Equions: Liner, Nonliner, Ordinry, Pril, Cmbridge Universiy Press, 003 [6] JC Bucher, The Numericl Mehods for Ordinry Differenil Equions, John Wiley nd Sons Ld, 003 [7] RM Mheij nd J Molenr, Ordinry Differenil Equions in Theory nd Prcice, SIAM, 00 [8] J Hubbrd nd B Wes, Differenil Equions: A Dynmicl Sysems Approch: Ordinry Differenil Equions, New York, Springer-Verlg, 00 [9] P Hrmn, Ordinry Differenil Equions, SIAM, 00 [0] E Hirer e l, Solving Ordinry Differenil Equions I, New York, Springer-Verlg, 987 [] HT Dvis, Inroducion o Nonliner Differenil nd Inegrl Equions, NewYork,DoverPublicions Inc, 96

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