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An autonomous linear system is given by dx = x dt dy dt ky with initial point (xo,Yo) where k is a nonzero constant. The above system has the solution, x(t) = xoe' and y(t) = yoekt which can be written as y = bxk where b = yo/x% is a positive constant. This autonomous linear system has the origin (0,0) as its critical point. Describe the behaviour of the critical point if (a) k = 1 (b) k > 1 (c) k < 0

An autonomous linear system is given by dx = x dt dy dt ky with initial point (xo,Yo) where k is a nonzero constant. The above system has the solution, x(t) = xoe' and y(t) = yoekt which can be written as y = bxk where b = yo/x% is a positive constant. This autonomous linear system has the origin (0,0) as its critical point. Describe the behaviour of the critical point if (a) k = 1 (b) k > 1 (c) k < 0

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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An autonomous linear system is given by dx = x dt dy dt ky with initial point (xo,Yo) where k is a nonzero constant. The above system has the solution, x(t) = xoe' and y(t) = yoekt which can be written as y = bxk where b = yo/x% is a positive constant. This autonomous linear system has the origin (0,0) as its critical point. Describe the behaviour of the critical point if (a) k = 1 (b) k > 1 (c) k < 0
Transcribed Image Text:An autonomous linear system is given by dx = x dt dy dt ky with initial point (xo,Yo) where k is a nonzero constant. The above system has the solution, x(t) = xoe' and y(t) = yoekt which can be written as y = bxk where b = yo/x% is a positive constant. This autonomous linear system has the origin (0,0) as its critical point. Describe the behaviour of the critical point if (a) k = 1 (b) k > 1 (c) k < 0
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