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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2004 • Question 4

Question

Math 6410 • ODE Winter 2004 • Question 4

Nonhomogeneous Equations. Let \(v_0,x_0\in \mathbb {R}^n\) and \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le -\epsilon _0\) for all \(i=1,\ldots ,n\) and for some \(\epsilon _0>0\). Show that solution of the initial value problem (2.) is bounded for \(t\ge 0\) as long as it exists. Argue that therefore it exists for all time. \[ \begin {cases} \dfrac {dx}{dt}=Ax+e^{-t}(x+v_0),\\ x(0)=x_0. \end {cases} \tag {2.} \]

Raw LaTeX

\textbf{Nonhomogeneous Equations.}
Let \(v_0,x_0\in\R^n\) and \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\)
satisfy \(\operatorname{Re}\lambda_i\le-\epsilon_0\) for all \(i=1,\ldots,n\) and for some
\(\epsilon_0>0\). Show that solution of the initial value problem (2.) is bounded for
\(t\ge0\) as long as it exists. Argue that therefore it exists for all time.
\[
\begin{cases}
  \dfrac{dx}{dt}=Ax+e^{-t}(x+v_0),\\
  x(0)=x_0.
\end{cases}
\tag{2.}
\]

Topic Tags

Distributions and Weak Solutions Existence and Uniqueness Theorems Poincare Maps and Periodic Solutions Spectral Theory of Operators