Figure $2$ shows a liquid$-$level system in which two tanks have cross$-$sectional areas A$1$ and A$2,$ respectively. A pump is connected to the bottom of tank $1$ through a valve of linear resistance R$1.$ The liquid flows from tank $1$ to tank $2$ through a valve of linear resistance R$2$ and leaves tank $2$ through a valve of linear resistance R$3.$ The density of the liquid is constant.

a$.$ Derive the differential equations in terms of the liquid heights h$1$ and h$2.$ Write the equations in second$-$order matrix form.

b$.$ Assume the pump pressure $$p as the input and the liquid heights h$1$ and h$2$ as the outputs. Determine the state$-$equations form of the system.