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3. Let (rt), o(t)) be the polar coordinates for (r(t),y(t)). Prove that if lim T(t) 0 then the critical point (0,0) is asymptotically stable.

3. Let (rt), o(t)) be the polar coordinates for (r(t),y(t)). Prove that if lim T(t) 0 then the critical point (0,0) is asymptotically stable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. Let (rt), o(t)) be the polar coordinates for (r(t),y(t)). Prove that if lim T(t) 0 then the critical point (0,0) is asymptotically stable.
Transcribed Image Text:3. Let (rt), o(t)) be the polar coordinates for (r(t),y(t)). Prove that if lim T(t) 0 then the critical point (0,0) is asymptotically stable.
Expert Solution
Step 1

To prove that the system is asymptotically stable under the given conditions.

Step 2

Note that (0,0) is a critical point of the system as both dx/dt and dy/dt=0 at (0,0)

Advanced Math homework question answer, Step 2, Image 1
Step 3

Definition of asymptotic stability of the system 

Advanced Math homework question answer, Step 3, Image 1
Step 4

So, we need to prove that the eigenvalues of the matrix are negative under the given condition that r(t) tends to 0 as t tends to infinity. 

Advanced Math homework question answer, Step 4, Image 1
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