I Question about using matrices for differential equations

EtherNohow
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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!
 
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Given a system of equations ##\dot{\mathbf{x}}(t) = A \mathbf{x}(t)##, what is the general solution of this problem?
 
micromass said:
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?
 
That doesn't answer my question at all.
 
micromass said:
That doesn't answer my question at all.
Wouldn't it just be A? Since other than that both sides are the same?
 
In your course, what does it tell you about systems of differential equations? What book are you reading?
 
micromass said:
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.
 
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