Chapter 3.1, Problem 19E

Convert the third-order differential equation

$ $\frac{d^{3}y}{d{t}^3}+p\frac{d^{2}y}{d{t}^2}+q\frac{dy}{dt}+ry=0$ $

where p, q, and $r$ are constants, to a three-dimensional linear system written in matrix form.

In Exercises $20-23,$ we consider the following model of the market for single-family housing in a community. Let $S(t)$ be the number of sellers at time t, and let $B(t)$ be the number of buyers at time t We assume that there are natural equilibrium levels of buyers and sellers (made up of people who retire, change job locations, or wish to move for family reasons). The equilibrium level of sellers is $S_0$ and the equilibrium level of buyers is $B_0$

However, market forces can entice people to buy or sell under various conditions. For example, if the price of a house is very high, then house owners are tempted to sell their homes. If prices are very low, extra buyers enter the market looking for bargains. We let $b(t)=B(t)-B_0$ denote the deviation of the number of buyers from equilibrium at time t. So if $b(t)>0,$ then there are more buyers than usual, and we say it is a "seller's market." Presumably the competition of the extra buyers for the same number of houses for sale will force the prices up (the law of supply and demand).

Similarly, we let $s(t)=S(t)-S_0$ denote the deviation of the number of sellers from the equilibrium level. If $s(t)>0$ , then there are more sellers on the market than usual; and if the number of buyers is low, there are too many houses on the market and prices decrease, which in turn affects decisions to buy or sell.

We can give a simple model of this situation as follows:
$\frac{dY}{dt}=AY=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\left(\begin{array}{l}b\hfill \\ s\hfill \end{array}\right),\text{ where }Y=\left(\begin{array}{l}b\hfill \\ s\hfill \end{array}\right)$
The exact values of the parameters $\alpha ,\beta ,\gamma ,$ and $\delta$ depend on the economy of a particular community. Nevertheless, if we assume that everybody wants to get a bargain when they are buying a house and to get top dollar when they are selling a house, then we can hope to predict whether the parameters are positive or negative even though we cannot predict their exact values.

Use the information given above to obtain information about the parameters $\alpha ,\beta$

$\gamma ,$ and $\delta .$ Be sure to justify your answers.