## anonymous 5 years ago Find a basis for the set of solutions of the given differential equation: x'=|0 1 | x |-1 -1|

1. anonymous

Have you seen laplace transforms yet?

2. anonymous

no

3. 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)

4. 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.

5. anonymous

oh and its c2 not c1 the second time

6. anonymous

We're not supposed to use eignvectors yet

7. anonymous

X=x1 x2

8. anonymous

I don't see why not.

9. anonymous

X'=[x2] [-x1-x2]

10. anonymous

hm? I know they x is a vector.

11. anonymous

I'm sorry I can't come up with another way to do this.

12. anonymous

its cool thanks anyways

13. 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.

14. anonymous

Are you using a textbook. Differential Equations and their applications?

15. 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.