I Question about using matrices for differential equations

1. Mar 31, 2016

EtherNohow

Let x(t)=
[x1(t)
x2(t)]
be a solution to the system of differential equations:

x′1(t)=−2x1(t)+2x2(t)
x′2(t)==−6x1(t)+9x2(t)

If x(0)=
[4
-2]
find x(t).

I got the eigenvalues to be -6 and -5, but I don't know how to calculate the coefficients in front of the exponents. For lambda=-6 I get the vector (1, -2) and for lambda=-5 I get the vector (2, -3). I think these would be the coefficients, but I'm not sure, and I don't know how to use the initial values for x(0). Thanks for your help!

2. Mar 31, 2016

micromass

Given a system of equations $\dot{\mathbf{x}}(t) = A \mathbf{x}(t)$, what is the general solution of this problem?

3. Mar 31, 2016

EtherNohow

I put the vectors eigenvectors from into a matrix and put 4 and -2 on the right and solved for the two variables. I got -8 and 6 but the website says only -8 is right, and in don't know where to get the other two coefficients?

4. Mar 31, 2016

micromass

That doesn't answer my question at all.

5. Mar 31, 2016

EtherNohow

Wouldn't it just be A? Since other than that both sides are the same?

6. Mar 31, 2016

micromass

In your course, what does it tell you about systems of differential equations? What book are you reading?

7. Mar 31, 2016

EtherNohow

I haven't taken differential equations yet. I'm in matrix algebra using Elementary Linear Algebra 7e by Larson.