Answer
$\begin{bmatrix}
x_1'\\
x_2'
\end{bmatrix}=\begin{bmatrix}
0 & -\sin t & 1\\
-e^t & 0 & t^2 \\
-t & t^2 & 0
\end{bmatrix}.\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}+\begin{bmatrix}
t\\
t^3\\
1
\end{bmatrix}$
Work Step by Step
The vector formulation for the given system of differential equations is:
$\begin{bmatrix}
x_1'\\
x_2'
\end{bmatrix}=\begin{bmatrix}
0 & -\sin t & 1\\
-e^t & 0 & t^2 \\
-t & t^2 & 0
\end{bmatrix}.\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}+\begin{bmatrix}
t\\
t^3\\
1
\end{bmatrix}$
