Find a basis for the set of solutions of the given differential equation:
x'=|0 1 | x
|-1 -1|

Question

Find a basis for the set of solutions of the given differential equation:
x'=|0 1   | x
     |-1 -1|

anonymous · Answer

Have you seen laplace transforms yet?

anonymous · Answer

no

anonymous · Answer

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)

anonymous · Answer

$$x(t) = c_1 e^{\lambda_1 t} e_1 + c_1 e^{\lambda_2 t} e_2$$ where e1 and e2 are the eigenvectors we found.

anonymous · Answer

oh and its c2 not c1 the second time

anonymous · Answer

We're not supposed to use eignvectors yet

anonymous · Answer

X=x1
    x2

anonymous · Answer

I don't see why not.

anonymous · Answer

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

anonymous · Answer

hm? I know they x is a vector.

anonymous · Answer

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

anonymous · Answer

its cool thanks anyways

anonymous · Answer

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 · Answer

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

anonymous · Answer

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.