What are the eigenvalues X₁ (t) and X₂(t) of A(t)?
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 \( y(t) \equiv 0 \) is asymptotically stable if for some \( \theta < 0 \), the eigenvalues \( \lambda_1(t) \) and \( \lambda_2(t) \) of \( A(t) \) satisfy
\[ \text{Re}(\lambda_1(t)), \text{Re}(\lambda_2(t)) \leq \theta \text{ for all } t \geq 0. \]
Unfortunately, **this is false.** Consider, for instance, the following system:
\[ y' = \begin{pmatrix} -1 & e^{2t} \\ 0 & -1 \end{pmatrix} y \]
What are the eigenvalues \( \lambda_1(t) \) and \( \lambda_2(t) \) of \( A(t) \)? Show that the steady state solution \( y(t) \equiv 0 \) is *unstable*: there are solutions that start arbitrarily close to \( y \equiv 0 \) and then move a fixed distance away.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea01508-f9b6-4a19-b038-56e12c298daf%2F9294c652-b34a-4f4b-9ab6-9cadd73bfc67%2Fzspyjs_processed.png&w=3840&q=75)
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 \( y(t) \equiv 0 \) is asymptotically stable if for some \( \theta < 0 \), the eigenvalues \( \lambda_1(t) \) and \( \lambda_2(t) \) of \( A(t) \) satisfy
\[ \text{Re}(\lambda_1(t)), \text{Re}(\lambda_2(t)) \leq \theta \text{ for all } t \geq 0. \]
Unfortunately, **this is false.** Consider, for instance, the following system:
\[ y' = \begin{pmatrix} -1 & e^{2t} \\ 0 & -1 \end{pmatrix} y \]
What are the eigenvalues \( \lambda_1(t) \) and \( \lambda_2(t) \) of \( A(t) \)? Show that the steady state solution \( y(t) \equiv 0 \) is *unstable*: there are solutions that start arbitrarily close to \( y \equiv 0 \) and then move a fixed distance away.
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