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solutions7bd

# solutions7bd - Math 33b Quiz 7bd Name UCLA ID Consider the...

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Math 33b, Quiz 7bd, March 12, 2008 Name: UCLA ID: Consider the system of differential equations x ( t ) = M x , where M is each of the matrices below. For each system of differential equations, a sample solution in the x - y plane is plotted below. Below each matrix, write the letter of the corresponding graph. (You need not show your work.) Do not assign more than one graph to a matrix or vice-versa. i. parenleftbigg - 0 . 1 - 0 . 7 0 . 6 - 0 . 1 parenrightbigg . ii. parenleftbigg - 0 . 3 0 . 1 0 . 5 0 . 1 parenrightbigg . iii. parenleftbigg - 0 . 4 0 . 2 - 0 . 4 0 . 2 parenrightbigg . iv. parenleftbigg - 0 . 2 - 0 . 1 0 . 1 0 parenrightbigg . (Note that in the actual quiz which was handed out in section, arrows were hand-drawn on each of the graphs. In A and B, the arrow points in towards the origin. In C and D, it points to the right.) -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 A -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 B -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 C -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 D Solution. i-B, ii-C, iii-D, iv-A. 1

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In i, Trace M = - 0 . 2 and det M = 0 . 43. Since (Trace M ) 2 - 4 det M < 0, the eigenvalues are complex conjugates. Since the trace is negative, the eigenvalues have negative real part.
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