Math 0250 Exam 2Name:

March 13, 1998

F. Beatrous

1.
Find all eigenvalues, and an eigenvector for each, for the matrix

\begin{displaymath}\left[\begin{array}{rr}0&a\\ a&0\end{array}\right]
\end{displaymath}

Assume that the number a is not 0.

2.
 The matrix

\begin{displaymath}A = \left[\begin{array}{rr}1&-1\\ -1&1\end{array}\right]
\end{displaymath}

has eigenvalues 0 and 2, with corresponding eigenvectors

\begin{displaymath}\left[\begin{array}{r}1\\ 1\end{array}\right] \quad\hbox{and}\quad
\left[\begin{array}{r}-1\\ 1\end{array}\right]
\end{displaymath}

respectively.
(a)
Give the general solution to the system X'=AX.

(b)
Find the fundamental matrix $\Psi(t)$ for the system X'=AX.
(c)
Find the solution to X'=AX satisfying the initial condition

\begin{displaymath}X(0) = \left[\begin{array}{r}1\\ 0\end{array}\right]
\end{displaymath}

3.
(a)
 With A as in problem 2, find a particular solution to the system

\begin{displaymath}X' = A X + \left[\begin{array}{r}1\\ -5\end{array}\right]e^{-t}
\end{displaymath}

(b)
Find the solution to the system in 3a satisfying the initial condition

\begin{displaymath}X(0) = \left[\begin{array}{r}0\\ 0\end{array}\right]
\end{displaymath}

(c)
With A as in problem 2, find a particular solution to

\begin{displaymath}X'= AX + \left[\begin{array}{r}1\\ 0\end{array}\right] e^t \cos t
\end{displaymath}

4.
Two 100 gallon tanks are connected by a system of pipes as indicated. Both tanks are initially filled with pure water. Starting at time t=0 minutes, fluid is pumped through the all the pipes at a rate of 10 gallons per minute in the directions indicated by the arrows. The fluid flowing into the left tank contains salt at a concentration of 10 pounds per gallon, while the fluid flowing in to the right tank contains 5 pounds of salt per gallon. Set up a system of differential equations and initial conditions for the amount of salt in each tank. What is the coefficient matrix for the system? What is the forcing function? Do not solve the system.

\includegraphics{tanks.ps}



 

Frank Beatrous
1998-10-26