Question 1 For the systems below, find and classify the critical points using linear stability method, also indicate if the equilibria are stable, asymptotically stable, or unstable. da dy (a) x² + y²-1, xy. dt dt

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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plz solve question 1(a) it within 30-40 mins I'll give you multiple upvote
Question 1.
For the systems below, find and classify the critical points using linear stability method,
also indicate if the equilibria are stable, asymptotically stable, or unstable.
da
dy
(a)
x² + y² -1,
xy.
dt
dt
da
dy
(b)
dt
ye", =y=x+y².
dt
da
dy
(c)
= x(2-x+y),
= y(4-x).
dt
dt
da
dy
(d)
=
= x(1-x+y),
= y(1-y + x).
dt
dt
da
dy
(e)
=-x+ray-x²,
= y(1-y), where r > 1.
dt
dt
du
du
(f)
u(v - 1),
-4-u² - v².
dt
dt
Question 2
Determine the stability property of the critical point at the origin for the following systems
(Try V(31, 32) yi + cy):
dy₁
dy₂
(a)
-yi +919²,
=-2yty2 - y₂.
dt
dt
dy₁
(b)
=
2y1y + 4y1y2 + 2y2.
dt
dy₁
dy2
(c)
-2y + 2y²,
-2y132.
dt
dt
Question 3
(a) Show that the critical point at the origin for the system
-x³ + xy²
-2x²y-y³
y'
=
is asymptotically stable (hint: V(x, y) = ax² + cy²).
- y₁ - y2¹
dy₂
dt
Transcribed Image Text:Question 1. For the systems below, find and classify the critical points using linear stability method, also indicate if the equilibria are stable, asymptotically stable, or unstable. da dy (a) x² + y² -1, xy. dt dt da dy (b) dt ye", =y=x+y². dt da dy (c) = x(2-x+y), = y(4-x). dt dt da dy (d) = = x(1-x+y), = y(1-y + x). dt dt da dy (e) =-x+ray-x², = y(1-y), where r > 1. dt dt du du (f) u(v - 1), -4-u² - v². dt dt Question 2 Determine the stability property of the critical point at the origin for the following systems (Try V(31, 32) yi + cy): dy₁ dy₂ (a) -yi +919², =-2yty2 - y₂. dt dt dy₁ (b) = 2y1y + 4y1y2 + 2y2. dt dy₁ dy2 (c) -2y + 2y², -2y132. dt dt Question 3 (a) Show that the critical point at the origin for the system -x³ + xy² -2x²y-y³ y' = is asymptotically stable (hint: V(x, y) = ax² + cy²). - y₁ - y2¹ dy₂ dt
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