Question about using matrices for differential equations

[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!

 

[micromass]

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

 

[EtherNohow]

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

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?

 

[micromass]

That doesn't answer my question at all.

 

[EtherNohow]

That doesn't answer my question at all.

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

 

[micromass]

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

 

[EtherNohow]

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

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