Consider the model system of three coupled, linear ODEs described by:

$2\frac{d{y}_1}{dt}=0\mathrm{.}5u_1(t)+1\mathrm{.}2u_2(t)-y_1(t)$

$\frac{d{y}_2}{dt}=2y_1(t)-y_2(t)$

$\frac{d{y}_3}{dt}=0\mathrm{.}75u_3(t)$

Assume that all six quantities $y_1(t),y_2(t),y_3(t)u_1(t),u_2(t)$ and $u_3(t)$ are perturbation variables. Show that there are five $($non$-$zero$)$ transfer functions, relating the three inputs $u_1(t),u_2(t)$ and $u_3(t)$ to the three outputs $y_1(t),y_2(t)$ and $y_3(t).$ Find these five transfer functions: $G_{11},G_{12},G_{21},G_{22},$ and $G_{33}$ by writing the model as:

$Y_1(s)=G_{11}(s)U_1(s)+G_{12}(s)U_2(s)$

$Y_2(s)=G_{21}(s)U_1(s)+G_{22}(s)U_2(s)$

$Y_3(s)=G_{33}(s)U_3(s)$

Also indicate if the five transfer function can be classified as integrator, first$-$order, or second order transfer functions.