Show me the steps of determine blue and inf is here Hence we obtain the linearized equation of Eq.(8) about its unique positiveequilibrium point y as follow:2pZn-m= 0.(15)Zn+1Therefore, the characteristic equation of Eq.(8) is2p0.тAm+1(16)YnYn+1 = 1++p(8)Yn-m..boneB.u. hendle theinen og uetion (0)The unigue Theorem 10 Every solution of Eq. (8) satisfies both of the following asymptoticrelationsYn+1limAjl,Yn – ỹn 00lim sup (lyn – y|)/"where j e {1,.. , k} and X; are the roots of characteristic equation (16).Proof. We get from Eq.(8):-9 - (1+)-(*)Yn1+P.2Yn+1n-mp(y + Yn-m)ý · Yn-m(Yn–m – 9) .(Yn – 9).2yn-mSet en = Yn – ỹ. Therefore we haveen+1 + Pnen + Inen-m = 0,wherep(y + Yn-m)InYn-m,2y· Yn-mPn,2Due to the equilibrium point j of Eq.(8) is globally asymptotically stable, weget2plim Pnlim qnHence, the limiting equation of Eq.(8) is the linearized equation (15). -

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Theorem 10 Every solution of Eq. (8) satisfies both of the following asymptotic relations Yn+1 lim n 00 Yn lim sup (lyn - y|)/" n 00 where je {1,... , k} and X; are the roots of characteristic equation (16). Proof. We get from Eq.(8): -- - (1+)-(1+) Yn Yn+1 Yn-m p(y + Yn-m) ý. Yn-m (Yn – 9) (Yn-m – 9) . | yn-m Set en = Yn - g. Therefore we have en+1 + Pnen + Anen-m 0, where p (y + Yn-m) , In Pn 2. Yn-m Due to the equilibrium point y of Eq.(8) is globally asymptotically stable, we get 2p lim Pn lim qn n 00 Hence, the limiting equation of Eq.(8) is the linearized equation (15). I

Theorem 10 Every solution of Eq. (8) satisfies both of the following asymptotic relations Yn+1 lim n 00 Yn lim sup (lyn - y|)/" n 00 where je {1,... , k} and X; are the roots of characteristic equation (16). Proof. We get from Eq.(8): -- - (1+)-(1+) Yn Yn+1 Yn-m p(y + Yn-m) ý. Yn-m (Yn – 9) (Yn-m – 9) . | yn-m Set en = Yn - g. Therefore we have en+1 + Pnen + Anen-m 0, where p (y + Yn-m) , In Pn 2. Yn-m Due to the equilibrium point y of Eq.(8) is globally asymptotically stable, we get 2p lim Pn lim qn n 00 Hence, the limiting equation of Eq.(8) is the linearized equation (15). I

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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