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What are the eigenvalues X₁ (t) and X₂(t) of A(t)?

What are the eigenvalues X₁ (t) and X₂(t) of A(t)?

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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Problem 5: Stability of solutions to linear systems with variable coefficients is much more complicated than the constant coefficient case. For instance, it might be tempting to think that for a system of the form y' = A(t)y the steady state solution y(t) = 0 is asymptotically stable if for some @ < 0, the eigenvalues X₁ (t) and X₂(t) of A(t) satisfy Re(A₁ (t)), Re(X₂(t)) ≤ 0 for all t≥ 0. Unfortunately, this is false. Consider, for instance, the following system: e2t y = 0 y What are the eigenvalues A₁ (t) and X₂(t) of A(t)? Show that the steady state solution y(t) = 0 is unstable: there are solutions that start arbitrarily close to y = 0 and then move a fixed distance away.
Transcribed Image Text:Problem 5: Stability of solutions to linear systems with variable coefficients is much more complicated than the constant coefficient case. For instance, it might be tempting to think that for a system of the form y' = A(t)y the steady state solution y(t) = 0 is asymptotically stable if for some @ < 0, the eigenvalues X₁ (t) and X₂(t) of A(t) satisfy Re(A₁ (t)), Re(X₂(t)) ≤ 0 for all t≥ 0. Unfortunately, this is false. Consider, for instance, the following system: e2t y = 0 y What are the eigenvalues A₁ (t) and X₂(t) of A(t)? Show that the steady state solution y(t) = 0 is unstable: there are solutions that start arbitrarily close to y = 0 and then move a fixed distance away.
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Step 1: Will Find the Eigen values of the given system

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