Here, we study the local stability character of the solutions of Eq.(1). Eq.(1) has a unique equilibrium point I = aT + or, T(1– a)(y+ 6) = 3a², if (1 – a)(y + 8) # B, then the unique equilibrium point is = 0. Let f: (0, 0)3 → (0, 00) be a function defined by Buv f(u, v, w) = av + (6) Yu + dw Accordingly, it follows that Bövw (yu + đw)2' Bu fu(u, v, w) fo(u, v, w) a + Yu + dw Bouv fw(u, v, w) (yu + dw)? Thus fu(T, F, T) -a2, (7 + 8)2 fo(T, T, T) a + %3D -38 fw (T, T, T) (y + 8)2 The linearized equation of Eq.(1) about is Yn+1 + a2Yn-i+ a1yn-2 + aoyn-4 = 0, Yn+1 + (y + 8)2 a + Y + 8 Yn + Уп-1 2 Yn-2 = 0. (7+ 6)
Here, we study the local stability character of the solutions of Eq.(1). Eq.(1) has a unique equilibrium point I = aT + or, T(1– a)(y+ 6) = 3a², if (1 – a)(y + 8) # B, then the unique equilibrium point is = 0. Let f: (0, 0)3 → (0, 00) be a function defined by Buv f(u, v, w) = av + (6) Yu + dw Accordingly, it follows that Bövw (yu + đw)2' Bu fu(u, v, w) fo(u, v, w) a + Yu + dw Bouv fw(u, v, w) (yu + dw)? Thus fu(T, F, T) -a2, (7 + 8)2 fo(T, T, T) a + %3D -38 fw (T, T, T) (y + 8)2 The linearized equation of Eq.(1) about is Yn+1 + a2Yn-i+ a1yn-2 + aoyn-4 = 0, Yn+1 + (y + 8)2 a + Y + 8 Yn + Уп-1 2 Yn-2 = 0. (7+ 6)
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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Transcribed Image Text:Bxn-10n-2
Xn+1 = axn-2+
п 3D 0, 1, ...,
(1)
YXn-1 + dxn-4
Transcribed Image Text:Here, we study the local stability character of the solutions of Eq.(1). Eq.(1) has a
unique equilibrium point
I = ax +
or,
a(1 – a)(y+ 8) = Bx²,
if (1 – a)(y+ 8) # B, then the unique equilibrium point is T = 0.
Let f : (0, 0)3
(0, 0) be a function defined by
Buv
f (u, v, w) :
(6)
= Qv +
YU + Sw
Accordingly, it follows that
Bövw
fu(u, v, w)
(yu + dw)2'
Ви
fo(u, v, w)
a +
Yu + Sw
-Bduv
fu (u, v, w)
(yu + dw)2
Thus
fu(T, T, T)
= -a2,
(y + 8)2
fo (T, T, T)
a +
= -a1,
fw (T, T, T)
(y + 8)2
The linearized equation of Eq.(1) about T is
Yn+1 + a2Yn-1+ a1Yn-2+ aoYn-4 = 0,
(7)
Уп+1 +
(y + 8)2 In
a +
Уп-1
(7 + 8)2 Yn-2 = 0.
5
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