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Theorem 1. Assume that B(y + 38) < (1 – a)(y+ 8)² and a< 1. - Then the equilibrium point of Eq.(1) is locally asymptotically stable. Proof: It is follows by Theorem A that, Eq.(7) is asymptotically stable if (y+8)2 + a + + (y+8)2 < 1,

Theorem 1. Assume that B(y + 38) < (1 – a)(y+ 8)² and a< 1. - Then the equilibrium point of Eq.(1) is locally asymptotically stable. Proof: It is follows by Theorem A that, Eq.(7) is asymptotically stable if (y+8)2 + a + + (y+8)2 < 1,

Advanced Engineering Mathematics
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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Theorem 1. Assume that B(y + 36) < (1 – a)(y + 8)² and a < 1. Then the equilibrium point of Eq.(1) is locally asymptotically stable. Proof: It is follows by Theorem A that, Eq.(7) is asymptotically stable if + a + < 1, (y + 8)²| + (y+ 8)² and so 3ô + a(y + 8)? + B(y+ 8) + 38 < (y + 8)². Therefore, 2,38 + 3(y+ 8) < (y + 8)² – a(y + 8)². Then 238 + By + Bồ < (1 – a)(y+ ô)². By + 3B8 (c+d)? < (1 — а). B(y+ 38) (y + 8)² < (1 – a). The proof is complete.
Transcribed Image Text:Theorem 1. Assume that B(y + 36) < (1 – a)(y + 8)² and a < 1. Then the equilibrium point of Eq.(1) is locally asymptotically stable. Proof: It is follows by Theorem A that, Eq.(7) is asymptotically stable if + a + < 1, (y + 8)²| + (y+ 8)² and so 3ô + a(y + 8)? + B(y+ 8) + 38 < (y + 8)². Therefore, 2,38 + 3(y+ 8) < (y + 8)² – a(y + 8)². Then 238 + By + Bồ < (1 – a)(y+ ô)². By + 3B8 (c+d)? < (1 — а). B(y+ 38) (y + 8)² < (1 – a). The proof is complete.
Bxn-10n-2 Xn+1 = axn-2 + n = 0, 1, ..., (1) YIn-1 + dxn-4 1. The linearized equation of Eq.-(1) about is Yn+1 + a2Yn-i+ a1Yn-2 + aoyn-4 = = 0, a + (7) Yn+1 + (7 + 6)2 Yn + Yn-1 (7+ 8)2 Yn-2 = 0.
Transcribed Image Text:Bxn-10n-2 Xn+1 = axn-2 + n = 0, 1, ..., (1) YIn-1 + dxn-4 1. The linearized equation of Eq.-(1) about is Yn+1 + a2Yn-i+ a1Yn-2 + aoyn-4 = = 0, a + (7) Yn+1 + (7 + 6)2 Yn + Yn-1 (7+ 8)2 Yn-2 = 0.
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