The system of differential equations is given as 1-a 1-a - [₁ i] a a-1 x': = x, where a is a constant parameter. (a) Find the critical value of a where the qualitative nature of the phase portrait changes. (b) Choose a value of a below the critical value and establish the general solution. What type of equilibrium (the origin point at the phase portrait) corresponds to the chosen a? (c) Choose a value of a above the critical value and establish the general solution. What type of equilibrium (the origin point at the phase portrait) corresponds to the chosen a? (d) For the critical value of a, what are the phase trajectories and how many equilibria are there?
The system of differential equations is given as 1-a 1-a - [₁ i] a a-1 x': = x, where a is a constant parameter. (a) Find the critical value of a where the qualitative nature of the phase portrait changes. (b) Choose a value of a below the critical value and establish the general solution. What type of equilibrium (the origin point at the phase portrait) corresponds to the chosen a? (c) Choose a value of a above the critical value and establish the general solution. What type of equilibrium (the origin point at the phase portrait) corresponds to the chosen a? (d) For the critical value of a, what are the phase trajectories and how many equilibria are there?
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:P5. The system of differential equations is given as
1 - α 1- α
[
a
x' =
α-
x,
where a is a constant parameter.
(a) Find the critical value of a where the qualitative nature of the phase portrait
changes.
(b) Choose a value of a below the critical value and establish the general solution.
What type of equilibrium (the origin point at the phase portrait) corresponds
to the chosen a?
(c) Choose a value of a above the critical value and establish the general solution.
What type of equilibrium (the origin point at the phase portrait) corresponds
to the chosen a?
(d) For the critical value of a, what are the phase trajectories and how many
equilibria are there?
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