Show me the steps of determine blue and all information is here Hence, the characteristic equation of (28) isP(1) = X® – n5 X5 – n4 X* – n3 X3 – n2 X² – n1 1 – no,(29)where,(2ε+6μ-2 εμ+ * 3)(2μ (1 - 9)(2ε+4μ- 6 εμ + 2 8μ - έ + 15)(2μ (ε - 1))(2 e + 3µ + 2 € µ – e² µ +2 e? – 12)(2 µ (e – 1))( ε + 2μ-2 εμ +3)(µ (1 – e))n5n4 =n3n2 =n1no = -1.Now, let T(A) = X° and V(A) = -n5 15 – n4 X4 – n3 X3 – n2 X² – nị 1 – no. Consider |A|=1,then one has(2ε+3μ + 2 εμ- *μ + 22 - 12)(2μ (ε= 1))( ε+2μ-2εμ 3)(µ (1 – €))(4ε + 7μ - 4εμ + ε' μ + ?+3 )(2μ (ε -1) )(2ε+6μ-2εμ+-3)(2μ (1 -6) )(2ε+4μ-6 εμ + 2 ε' μ - 2 + 15)(2 µ (e – 1))(2ε+6μ-2εμ + *-3)(2μ (1 - )(+2μ-2 εμ +3)(и (1 — е))(2 e + µ – 2e u +e² µ + 6)(2 µ (e – 1))(-3µ+2 € µ+ e² µ), H(e – 1) (1– €)(2μ (ε-1) )-3µ +2 €µ+ e² µ – 1µ (e – 1)²2 µ (e – 1)|V(A)|(1 – e)+1,2(1 – e)+ 1,2(ε+2μ-2 εμ+ 3)(и (1 — е))(1 – e)+1,2+1,µ(e – 1)-4μ+4με+1=+1,2μ (ε- 1)3 > 1.=Then, from Rouché's Theorem 1.2, V(A) and Y(A) + ¥ (A) = P(X) have the same number ofzeros in an open unit disk JA| < 1, that means there are only five roots lie inside unit disk.Thus, from Theorem 1.1 the equilibrium point Eo is not asymptotically stable. By solving (29),we find that A = 1, implies E, is nonhyperbolic point. In this paper, we solve and study the properties of the following systemÉ Wn-PE zn-hWn-pZn-hh=1p=0p=1h=0Wn+1andZn+1+E,(4)Zn - €Wn - Hwhere u and e are arbitrary positive real numbers with initial conditions w; and z; for i =-2, –1,0.The nontrivial positive equilibrium point of (4) is Eo = (w, z) = (A, 1) such that e > 1.e+3T((Wn + Wn-1) (žn-1 + Zn–2)wn-1+Wn-2) (2n + 2n-1)Ln+1+ µ, Wn, Wn-1,+ €, Zn, Zn–1Zn - EWn - µTheorem 4.1. The positive equilibrium point E, of system (4) is not asymptotically stableand nonhyperbolic point .Proof. The linearized system of (4) evaluated at Eo is written in the matrix form asLn+1Q Ln,where Ln = (wn, Zn, Wn–1, Zn–1, Wn-2, Zn-2)" and the Jacobian matrix determined at Eo is equal-(4µ)(e2+2 €-3)1-e2-2금e+3e+3-(e2+2€-3)-(e+3)2 µ-(e+3)2 µ1-€21(28)1119.

Note:- The characteristic polynomial of a matrix A is determinant of A-I x Here, I is identity…

Answered: Show me the steps of determine blue and…

Show me the steps of determine blue and all information is here

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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