Math 6610 • Analysis of Numerical Methods I

{Numerical ODEs}

\textbf{a)} The difference equation
\[
  y_{n+k}+a_1y_{n+k-1}+\cdots+a_ky_n=0
\]
can, after the substitution
\(\mathbf{y}_j=(y_j,y_{j+1},\ldots,y_{j+k-1})^T\), be written in the form
\(\mathbf{y}_{n+1}=A\mathbf{y}_n\). What does the matrix \(A\) look like? Describe the
condition for \(y\) to remain bounded in terms of \(A\).

\textbf{b)} The differential equation
\[
  y^{(k)}+a_1y^{(k-1)}+\cdots+a_ky=0
\]
can, after the substitution \(\mathbf{y}=(y,y',\ldots,y^{(k-1)})^T\), be written in the
form \(\mathbf{y}'=A\mathbf{y}\). What does the matrix \(A\) look like? Describe the
condition for \(y\) to remain bounded in terms of \(A\).

\textbf{c)} Show that all solutions of the difference equation
\[
  y_{n+1}-2\lambda y_n+y_{n-1}=0
\]
are bounded, as \(n\to\infty\), if \(\lambda\) is real and \(\lvert\lambda\rvert<1\).