anonymous
  • anonymous
Find a basis for the set of solutions of the given differential equation: x'=|0 1 | x |-1 -1|
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
Have you seen laplace transforms yet?
anonymous
  • anonymous
no
anonymous
  • anonymous
Find the eigenvalues of the matrix. Use the characteristic polynomial. Use the eigenvalues to find the eigenvectors. (I am assuming you know linear algebra, if you are in a differential course)

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anonymous
  • anonymous
\[x(t) = c_1 e^{\lambda_1 t} e_1 + c_1 e^{\lambda_2 t} e_2\] where e1 and e2 are the eigenvectors we found.
anonymous
  • anonymous
oh and its c2 not c1 the second time
anonymous
  • anonymous
We're not supposed to use eignvectors yet
anonymous
  • anonymous
X=x1 x2
anonymous
  • anonymous
I don't see why not.
anonymous
  • anonymous
X'=[x2] [-x1-x2]
anonymous
  • anonymous
hm? I know they x is a vector.
anonymous
  • anonymous
I'm sorry I can't come up with another way to do this.
anonymous
  • anonymous
its cool thanks anyways
anonymous
  • anonymous
the basis for the matrix is (1,0) (0,1) I don't know how to relate it to the solution which is of the form I gave above. But you NEED to know the eigenvalues. e1 and e2 can be any ol' basis vectors.
anonymous
  • anonymous
Are you using a textbook. Differential Equations and their applications?
anonymous
  • anonymous
Page 294 has an example really similar to yours. I can't see the entire book online. It's on google books, if you don't have a hard copy.

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