Tough System of Linear Differential Equations

[verdelaurence]

Using eigen values-eigen vector method to solve for vector(y) with the undetermine coefficients.
vector(y)'=(A)vector(y)+vector(g(t))
vector(y(0))=[0
0
0]
vector(y)=[y1
y2
y3]
A=|8 12 -2|
|-3 -4 1|
|-1 -2 2|
vctor(g(t))=e^(-3t)[3
2
4]

I think I got the eigen value and there was only one which was 2 but not sure where to go from there. Please help.




2 Also how would i solve the same problem with variation of parameters to solve for vector(y(t))

3 How would i solve same problem with Laplaces Transform to solve for vector(y)