1. (a) Find the general solution of the system y = Ay, where
1 2
4
3
A=
The characteristic polynomial is given by
2 2 + 5
The eigenvalues are = 1 + 2i and = 1 2i. The eigenvector, w, can be found from
1
(A I )w = 0. One such eigenvector is w =
. We can wr
MIDTERM 2
Math 33B
2/17/2010
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Solutions to Hour Exam I
The problems on this exam came in multiple versions and were scrambled, so I
have just labelled them A, B, C,. here. The problems from your exam are here
somewhere.
A. Consider the dierential equation y = f (t, y ), where f (t, y
1.
(15 points ) Give general solutions to the following dierential equations:
x2
(a)
z = cos z
dz
x2
The equation dx = cos z is separable. We first separate variables and then integrate,
3
cos z dz = x2 dx. This gives sin z = x + C .
3
(b)
tu = eu
The equ