I am asked to find the solution of the following system of nonhomogeneus differential equations using elimination method:
$$ \begin{cases} x'(t)=4x(t)-2y(t)+t^{-3} \\ y'(t)=8x(t)-4y(t)-t^{-2} \end{cases} $$
By differentiating the first equation and substituing in values from second one, I find the characteristic value which is $ \lambda_{1,2} = 0 $, but from there on I am not sure how to proceed.
I believe the solution of the homogeneus equation for $x(t)$ is $ x_{0}(t) = C_1e^{0t} = C_1 $, but I do not know in what form should I seek a particular solution. Some help or hints on what to do would be greatly appreciated. Thank you.