Differential equation to state space representation
You have
$$ a \, \dot{x}_1 + b \, \dot{x}_2 + c \,x_1 + d \, x_2 + e = u_1 \\
f \, \dot{x}_1 + g \, \dot{x}_2 + h \,x_1 + i \,x_2 + j = u_2 $$
That can be re written as
$$ a \, \dot{x}_1 + b \, \dot{x}_2 + c \,x_1 + d \, x_2 = u_1-e \\
f \, \dot{x}_1 + g \, \dot{x}_2 + h \,x_1 + i \,x_2 = u_2-j $$
And then
$$
H = \begin{bmatrix} a &amp; b \\f &amp; g\end{bmatrix}, \quad -A = \begin{bmatrix} c &amp; d \\h &amp; i\end{bmatrix}, \quad \tilde{u} = \begin{bmatrix} u_1 -e \\ u_2 -j \end{bmatrix} \text{ and } x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}.\\
H \dot{x} - Ax = \tilde{u},\\
H \dot{x} = Ax + \tilde{u}. 
$$
That representation should be somewhat familiar to you except that it is a descriptor representation due to the \$H\$ matrix. If that matrix is invertible you will be able to find the usual \$\dot{x} = Ax+Bu\$. But if it is not invertible then you should dig into descriptor representation theory and find out when you can find a state-space representation to them. Or read about Differential-algebraic system of equations. I guess that \$H\$ having a left-inverse would be a must, but I don't know a thing about the descriptor representation.