status equations of 2° order circuit
Starting from your equations:

$$\begin{cases}
\frac{\text{d}}{\text{d}t}i_1=-\frac{R_1}{L_1+L_2} i_1+\frac{L_2}{L_1+L_2}\frac{\text{d}}{\text{d}t}i_2\\
\frac{\text{d}}{\text{d}t}i_2=-\frac{R_2}{L_2}i_2+\frac{\text{d}}{\text{d}t}i_1+e \\
\end{cases}$$

Substitution of second equation into first equation, and of first equation into second:

$$\begin{cases}
\frac{\text{d}}{\text{d}t}i_1=-\frac{R_1}{L_1+L_2} i_1+\frac{L_2}{L_1+L_2}\left(-\frac{R_2}{L_2}i_2+\frac{\text{d}}{\text{d}t}i_1+e\right) \\
\frac{\text{d}}{\text{d}t}i_2=-\frac{R_2}{L_2}i_2+\left(-\frac{R_1}{L_1+L_2} i_1+\frac{L_2}{L_1+L_2}\frac{\text{d}}{\text{d}t}i_2\right)+e \\
\end{cases}$$

Rewriting equations:

$$\begin{cases}
\frac{\text{d}}{\text{d}t}i_1\times\left(1-\frac{L_2}{L_1+L_2}\right)= -\frac{R_1}{L_1+L_2}i_1-\frac{R_2}{L_1+L_2}i_2+\frac{L_2}{L_1+L_2}e\\
\frac{\text{d}}{\text{d}t}i_2\times\left(1-\frac{L_2}{L_1+L_2}\right)= -\frac{R_1}{L_1+L_2}i_1 -\frac{R_2}{L_2}i_2 + e\\
\end{cases}$$

$$\begin{cases}
\frac{\text{d}}{\text{d}t}i_1\times L_1= -R_1i_1-R_2i_2+L_2e\\
\frac{\text{d}}{\text{d}t}i_2\times L_1= -R_1i_1 -\frac{(L_1+L_2)R_2}{L_2}i_2 + (L_1+L_2)e\\
\end{cases}$$

Matrix representation:

$$
A = \frac{1}{L_1}\left( \begin{array}{cc}
-R_1 &amp; -R_2 \\ -R_1 &amp; -\frac{(L_1+L_2)R_2}{L_2} \end{array} \right)
$$

$$
B = \frac{1}{L_1}\left( \begin{array}{c}
L_2 \\ L_1+L_2 \end{array} \right)
$$