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Math 0250 Final Examination


April 23, 1998


10:00 to 11:50 am





Instructions

1.
Before you begin, enter your name and student id number in the spaces below, and circle the name of your instructor.

2.
Show all your work on the exam itself. If you need additional space, use the backs of the pages.

3.
You may not use books or notes on the exam.

4.
You may use a calculator, but calculators may not be shared.


Name  
Student ID number  


Instructor (circle one): Beatrous    Hart    Marsden




1 12                 
2 10  
3a 12  
3b 12  
4 12  
5a 12  
5b 12  
6 12  
7a 12  
7b 12  
8a 12  
8b 12  
8c 12  
8d 12  
9a 12  
9b 12  
10 10  
Total 200  

1.
 The complex number i has three cube roots. Find them.
2.
 Find the inverse of the matrix

\begin{displaymath}\left[\begin{array}{rrr}1&-2&-1\\ 0&1&1\\ -1&1&1\end{array}\right]
\end{displaymath}

3.
Solve the following:
(a)
  $t u' + u = 2t, \quad u(1) = 3$
(b)
  $t u' - u^2 = 1, \quad u(1) = 0$

4.
 Find the eigenvalues and corresponding eigenvectors for the matrix

\begin{displaymath}\left[\begin{array}{cc}a&1\\ a^2&a\end{array}\right]
\end{displaymath}

5.
(a)
 The matrix

\begin{displaymath}A= \left[\begin{array}{rr}1&1\\ 4&1\end{array}\right]
\end{displaymath}

has eigenvalues 3 and -1 with corresponding eigenvectors

\begin{displaymath}\left[\begin{array}{r}1\\ 2\end{array}\right] \quad\hbox{and}\quad
\left[\begin{array}{r}1\\ -2\end{array}\right]
\end{displaymath}

Solve the initial value problem

\begin{displaymath}X' = A X
\qquad X(0) = \left[\begin{array}{r}2\\ 3\end{array}\right]
\end{displaymath}

(b)
  The matrix

\begin{displaymath}A = \left[\begin{array}{rr}1&-1\\ 5&-3\end{array}\right]
\end{displaymath}

has eigenvalues -1+i and -1-i with corresponding eigenvectors

\begin{displaymath}\left[\begin{array}{c}5\\ 2+i\end{array}\right]\quad\hbox{and}\quad
\left[\begin{array}{c}5\\ 2-i\end{array}\right]
\end{displaymath}

Solve the initial value problem

\begin{displaymath}X' = A X
\qquad X(0) = \left[\begin{array}{r}1\\ 2\end{array}\right]
\end{displaymath}

6.
 Find a particular solution to

\begin{displaymath}X' = \left[\begin{array}{rrr}1&1&1\\ 0&3&2\\ 0&0&5\end{array}...
... X
+ \left[\begin{array}{r}1\\ 0\\ 0\end{array}\right] \cos t
\end{displaymath}

7.
The functions

\begin{eqnarray*}u_1(t) &=& t\\ u_2(t) &=& e^t.
\end{eqnarray*}


are solutions to the homogeneous linear differential equation

(1-t) u'' + t u' - u = 0

(a)
 Show that u1 and u2 form a basic set of solutions on the interval -1<t<1.
(b)
  Find the general solution of the non-homogeneous system

(1-t) u'' + t u' -u = 4.

Hint: Look for a constant solution.

8.
Find a general solution for each of the following differential equations
(a)
  u''+7u'+6u = 0
(b)
  u''+6u'+13u=0
(c)
  u''+12u'+36u=18
(d)
  u(4) - u = t

9.
A damped mass spring system is periodically forced, so that it oscillates according to the equation

\begin{displaymath}y'' + 2 y' + 5y = 2 \cos (2\pi t)
\end{displaymath}

(a)
 Give a convincing argument that the system eventually settles into a periodic oscillation. (Note: The reader is more likely to be convinced by an argument based on mathematical principles than one based on physical intuition.)

(b)
 Calculate the frequency and amplitude of the eventual oscillation.

10.
 Give a first order system, with initial condition, that's equivalent to

\begin{eqnarray*}y''' - 2y'' + y' - y &=& \cos t\\
y(0) &=& 1\\
y'(0) &=& 2\\
y''(0) &=& 3
\end{eqnarray*}



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Frank Beatrous
1998-12-03