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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1995 • Question 3

Question

Math 6410 • ODE Fall 1995 • Question 3

(Concrete cases from 2.) Change to vector-matrix form and discuss the stability of the trivial solution of the following equations.

  1. \(u''-u=0\)
  2. \(u''+2ku'+u=0\) with \(k>0\)
  3. \(u^{(4)}+2u''+u=0\)
  4. \(y'=\Theta y\) where \(\Theta \) is the matrix of all zeros.

Raw LaTeX

(Concrete cases from 2.) Change to vector-matrix form and discuss the stability of
the trivial solution of the following equations.
\begin{enumerate}
\item[(i)] \(u''-u=0\)
\item[(ii)] \(u''+2ku'+u=0\) with \(k>0\)
\item[(iii)] \(u^{(4)}+2u''+u=0\)
\item[(iv)] \(y'=\Theta y\) where \(\Theta\) is the matrix of all zeros.
\end{enumerate}

Topic Tags

Asymptotic and Exponential Stability Lyapunov Stability and Lyapunov Functions Periodic Orbits, Limit Cycles, and Orbital Stability Stability of Linear Systems