I want to convert the following 1st order non homogeneous differential equation system into 2nd order differential equation.
$\vec{x}'=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$
So i get $x'_1=x_1+2x_2+2t$ and $x'_2=3x_1+2x_2-4t$
After doing some calculations, I get the following differential equation $x''_1-3x'_1-4x_1-10t=0$ Is this correct? I want to verify it by checking general and particular solution to this 2nd order differential equation. I already have general and particular solution to 1st order differential equation system
We have
$$x' = x + 2 y + 2 t \\ y' = 3 x + 2 y - 4t$$
From the first equation, we have
$$x'' = x' + 2 y' + 2 = x' + 2(3x + 2 y - 4t)+ 2 = x' + 2(3x + (x'-x-2t) - 4t)+2$$
Simplifying
$$x'' - 3x' - 4x + 12 t - 2 = 0$$