Homework  7  427K-H  due Thursday March 27

Remember there is an exam on March 28 in the lecture room
.
1.  Find the general solution to the equation Y' = AY  where A is
the 3x3 matrix [1 0 5;1 1 3;0 2 1].


2.  If X' = AX  give the eigenvalues of A and determine the stability
of the fixed or equilibrium point 0  when:



a) A= [ 1, 0, 2;2,2,0;5,-2, 3].

b) A = 1,0,1,4;4,-1,0,3;3,-4,0,6;1,0,0,10].

c)  A = [1,-3;-2,10]

d) A = [-2,1,0;-5,-1,0; 1,0,3].

3. Find the general soution of the equation y' = Ay  where A is the

matrix  [-3, 3;-1, -5]

4. Find the general solution to the equation y' = Ay where A is the 

3x3 matrix [2 , 0 , 0;1 , 5 , 2; 3, -2, 5].  

5.Consider the 2x2 system of equations y' = Ay  where the matrix A

depends on an unknown parameter b.  A = [-1,b; -1, 1].  Classify the

solution  [0;0] as a fixed point for the various values  of b. Make

specific note of the value of b which is the transition point between

the different behaviors of solutions to y' = Ay.