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rocketboy
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Homework Statement
This is the second part of a multi-part question. Part (a) shows that:
x'' = Ax = [tex]\left(\stackrel{-2}{4/3}\stackrel{3/2}{-3}\right)[/tex]x
Part (b): Assume x = [tex]\epsilon[/tex]e[tex]^{rt}[/tex] and show that (A - r[tex]^{2}[/tex]I)[tex]\epsilon[/tex] = 0
x is the solution to the second order differential equation above, and [tex]\epsilon[/tex] is an eigenvector corresponding to the eigenvalue r[tex]^{2}[/tex] of A.
The Attempt at a Solution
Part (b): Alright, so given the above, I stated that [tex]\epsilon[/tex] = e[tex]^{-rt}[/tex]x
I then substituted everything into the left side of the equation I'm trying to prove to obtain:
(A - r[tex]^{2}[/tex]I)[tex]\epsilon[/tex] = [tex]\left(\stackrel{-2-r^{2}}{4/3}\stackrel{3/2}{-3-r^{2}}\right)[/tex][tex]\left(\stackrel{x_{1}}{x_{2}}\right)[/tex]e[tex]^{-rt}[/tex]
from here, I can see that I am not going in the right direction... any suggestions to get me moving along?
Thank-you!
-J
EDIT: for clarification, the above are 2x2 matrices... latex put the entries fairly close together. The matrix A has entries (-2 3/2) on the top and (4/3 -3) on the bottom.
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