Homework 3 Solution

# You are not allowed to scale the eigenvalues so you

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Unformatted text preview: igenvalues, wrote them as a + bi and pulled out common real factors. You are not allowed to scale the eigenvalues so you are stuck with the 1⁄2. For the eigenvectors, I wrote them as a + bi and eliminated fractions by scaling the vectors. You are allowed to scale eigenvectors. b) Note that Re(λ1 ) = 1 > 0 so this is a source spiral (or spiral source). 2 To determine the direction of the spin, notice that when (x, y ) = (1, 0) we have y (t) = 5 so we are moving in an upward direction at (1, 0). That means that the spiral must turn counterclockwise. c) spiral source (or source spiral) d) see b) e) √ √ √ √ −3 + 11i −3 − 11i 1/2(1+ 11i)t 1/2(1− 11i)t x(t) = c1 e + c2 e 10 10 —4— Winter 2014 Math 377 Homework III Solution f) Now things get messy. We just work on the ﬁrst solution (the c1 part) since all the information is buried in either of the solutions. √ √ √ 11t 11t −3 11 t/2 + i sin +i x1 (t) = e cos 10 0 2 2 √ √ √ 11t 11t −3 11 t/2 cos − sin Re(x1 (t)) = e 10 0 2 2 √ √ √ 11t 11t −3 11 t/2 sin Im(x1 (t)) = e + cos 10 0 2 2 √ t/2 x(t) = c1 e cos t/2 + c2 e 11t 2 √ sin −3...
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