I.  All parts of this question refer to the system
$$X' = \pmatrix{-2 & a\cr -2 & 0\cr}X\leqno(1)$$
II.  Consider the following initial value problem:
$$\cases{y'' - 6y' + 5y = 3e^x + 2\sin(x)&\cr
         y(0) = 0\cr
         y'(0) = 1\cr}\leqno(2)$$
\item{A)} (10) Find the general solution of the associated homogeneous
equation.
\item{B)} (10) Find a particular solution by the method of undetermined
coefficients
\item{C)} (10) Use parts A and B to find a solution of (2).  
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IV.  (15)  For which value of $\omega$ will {\it resonance} occur
in solutions of $y'' + 900y = A\cos(\omega t)$?
Find the general solution in the resonant case.
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V.  All parts of this question refer to the
autonomous first order system
$$\cases{{dx\over dt} = x - xy&\cr
         {dy\over dt} = y + 2xy&\cr}\leqno(3)$$
\item{A)} (7.5) Find all critical points of the system (3).
\item{B)} (5) The following is a Maple plot of
a direction field and trajectories for some autonomous
first order system.  
\vglue 5.0truein
Can this be the system
from (3)?  Explain how you can tell.
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