Math 277 01,

the course of Dr. Mihailovs

Midterm 2

October 28, 1998

  1. Find the eigenvalues and corresponding eigenvectors for the symmetric matrix $ A=\begin{pmatrix}-1 & 2\\ 2 & 2 \end{pmatrix}$.

  2. With $ A$ as in Problem 1, find an orthogonal matrix $ B$ such that $ B^{-1}AB$ is diagonal.

  3. Solve the initial-value problem $ \binom{\Dot{x}}{\Dot{y}}=\binom{-x+2y}{2x+2y}$, $ \binom{x}{y}(0)=\binom{3}{1}$ .

  4. Let $ A=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}$. Write down $ e^{At}$ (just give the answer).

  5. Let $ A=\begin{pmatrix}2 & 1\\ -1 & 2 \end{pmatrix}$. Write down the general real solution to $ \Dot{\bf x}=A{\bf x}$.

  6. By introducing a new variable $ y$, write $ \Ddot{x}-2\Dot{x}+x=0$ as a system of two first order linear equations of the form $ \Dot{\bf x}=A{\bf x}$.

  7. $ A=\begin{pmatrix}3 & 4\\ -1 &-1\end{pmatrix}$. Solve the initial-value problem $ \Dot{\bf x}=A{\bf x}$, $ {\bf x}(0)=\binom{3}{2}$.

  8. $ {\bf0}$ is an equilibrium point for the autonomous system $ \Dot{x}=-\sin(x+y),\thickspace \Dot{y}=xy+2x-4y$. Determine whether this equilibrium point is stable or unstable.

  9. Solve the initial-value problem $ y'=\frac{3x^2\cos y+y^2\cos x}{x^3\sin y-2y\sin x}$, $ y(0)=1$.

Copyright © 1998 Aleksandrs Mihailovs