Assume that the rise in the price `p=p(t)` of a product is proportional to the difference between the demand w(t) and the supply s(t) and that the demand depends on the price as a first degree polynomial. Set up a differential equation for the price.

Answer

To set up a differential equation for the price \( p(t) \) based on the given conditions, we can follow these steps:

### Step 1: Define the relationship between price, demand, and supply
According to the problem, the rise in price \( \frac{dp}{dt} \) is proportional to the difference between demand \( w(t) \) and supply \( s(t) \). We can express this relationship mathematically as:

\[ \frac{dp}{dt} = k(w(t) - s(t)) \]

where \( k \) is a constant of proportionality.

### Step 2: Express demand as a function of price
The problem states that the demand \( w(t) \) depends on the price \( p(t) \) as a first-degree polynomial. We can express this as:

\[ w(t) = ap + b \]

where \( a \) and \( b \) are constants.

### Step 3: Substitute the expression for demand into the differential equation
Now, we substitute the expression for demand \( w(t) \) into the differential equation we established in Step 1:

\[ \frac{dp}{dt} = k((ap + b) - s(t)) \]

### Step 4: Rearrange the equation
This can be rearranged to form a more standard differential equation:

\[ \frac{dp}{dt} = k(ap + b - s(t)) \]

### Final Form of the Differential Equation
Thus, the final differential equation that describes the rise in price \( p(t) \) is:

\[ \frac{dp}{dt} = k(ap + b - s(t)) \]