As there are constants and exponentials in this first-order linear ODE, we will use undetermined coefficients, postulating that the particular solution is
so that
Substituting our solution into the system we get,
This must equal and so we have
This gives us the system of equations
(based on the terms)
and
(based on comparing the constant vectors).
Looking at the first equation of the first system of equations, we have so that in the second equation,
Using the first equation again, we get
Looking at the first equation of the second system of equations, we have . From the second equation we get
and hence
after substituting back into the first equation.
The general solution to the ODE system will be where is our homogeneous solution from part (b).
Thus,
If
then
The first component tells us that while using this value of b in the second component gives .
Therefore, the solution is