Differential Equations

First Order Equations

1st Order Diff Equations Logo

Advertising Awareness

Solved Problems

Example 1.

Management of a company decided to advertise a new product permanently during the year. The advertising budget is $12,000. The coefficients \(k\) and \(b\) are \(k = {\frac{1}{4}},\;b = 25.\) Construct and solve the differential equation describing the number of people \(A\left( t \right)\) aware of this product.

Solution.

The equation of dynamics of \(A\left( t \right)\) is written in the form

\[\frac{{dA}}{{dt}} + kA = bq\left( t \right).\]

We assume that the time \(t\) is measured in months. By the condition of the problem, the advertising costs are constant during the year, so every month they are equal to

\[{q_0} = \frac{{12,000}}{{12}} = 1,000\,\frac{$}{\text{month}}.\]

Substituting the known data leads to the following differential equation:

\[\frac{{dA}}{{dt}} + \frac{A}{4} = 25,000.\]

The integrating factor for this equation has the form:

\[u\left( t \right) = {e^{\int {{\frac{1}{4}}dt} }} = {e^{\frac{t}{4}}}.\]

The general solution is written as

\[A\left( t \right) = \frac{{25,000\int {{e^{\frac{t}{4}}}dt} + C}}{{{e^{\frac{t}{4}}}}} = \frac{{\frac{{25,000}}{{\frac{1}{4}}}{e^{\frac{t}{4}}} + C}}{{{e^{\frac{t}{4}}}}} = 100,000 + C{e^{ - \frac{t}{4}}}.\]

We determine the constant of integration \(C\) from the initial condition \(A\left( {t = 0} \right) = 0.\) Hence, \(C = -100,000.\) Then the particular solution is expressed by the formula

\[A\left( t \right) = 100,000\left( {1 - {e^{ - \frac{t}{4}}}} \right).\]

The graph of this function is shown in Figure \(2\).

The number of potential buyers depending on time
Figure 2.

Hence in case of permanent advertising, the number of potential buyers aware of the product grows non-linearly approaching the maximum value

\[{A_{\max }} = \frac{{b{q_0}}}{k} = 100000.\]

Example 2.

Under the conditions of the previous problem \(1,\) figure out how will the number of potential buyers vary by the end of the year if the advertising budget is spent evenly for the first \(6\) months?

Solution.

In this problem the advertising mode has a stepwise shape. The advertising costs are shown schematically in Figure \(3\) below. We will investigate how awareness of consumers \(A\left( t \right)\) changes compared with the case of constant advertising over all year considered in Example \(1.\)

The problem is divided into two stages. The value of \(A\) is easily calculated by the end of sixth month using the formula

\[A\left( t \right) = \frac{{b{q_0}}}{k}\left( {1 - {e^{ - kt}}} \right),\]

derived in the previous Example. The coefficients have the values: \(k = {\frac{1}{4}},\) \(b = 25,\) \({q_0} = 2,000.\) Then

\[A\left( t \right) = 200,000\left( {1 - {e^{ - \frac{t}{4}}}} \right).\]

At the moment \(t = 6,\) we find that the number of consumers aware of the new product is

\[A\left( {t = 6} \right) = 200,000\left( {1 - {e^{ - \frac{6}{4}}}} \right) = 155,374.\]

In the second phase from the \(7\)th to the \(12\)th month, the advertising is completely absent. As a result, the level of awareness \(A\left( t \right)\) decreases according to the equation

\[\frac{{dA}}{{dt}} + kA = 0.\]

The solution of the homogeneous equation is the exponential function:

\[A\left( t \right) = C{e^{ - k\left( {t - 6} \right)}},\]

where \(t \gt 6\) months, and the constant \(C\) can be found from the initial condition for the \(2\)nd phase:

\[A\left( {t = 6} \right) = C{e^0} = C = 155,374 \approx 155,400.\]

Thus, the law \(A\left( t \right)\) in the second half-year is given by the formula

\[A\left( t \right) = 155,400\,{e^{ - \frac{{t - 6}}{4}}}.\]

Hence, the complete solution of the problem is written in the form:

\[A\left( {t} \right) = \begin{cases} {200,000\left( {1 - {e^{ - \frac{t}{4}}}} \right),\; 0 \le t \le 6 }\\ {155,400\,{e^{ - \frac{{t - 6}}{4}}},\; 6 \lt t \le 12} \end{cases}\]

The graph of the function \(A\left( t \right)\) is given in Figure \(4.\)

Stepwise law of advertising
Figure 3.
Awareness of consumers for homogeneous (example 1) and stepwise (example 2) advertising
Figure 4.

For comparison, the curve \(A\left( t \right)\) from the previous Example \(1\) is also shown on this chart. As it can be seen, awareness of consumers by the end of the year in the second case will be lower than in the case of homogeneous advertising. The exact values of \(A\) for both these cases are

\[\text{End of the year:}\;\;\;A\left( {\text{example}\,1} \right) = 95,021;\;\;\; A\left( {\text{example}\,2} \right) = 34,669.\]

Interestingly, that the average value of \(A\) over the year is greater for the second mode:

\[\text{Mean values:}\;\;\; \overline A\left( {\text{example}\,1} \right) = 72,117;\;\;\; \overline A\left( {\text{example}\,2} \right) = 89,825.\]

One can roughly assume that the volume of sales is proportional to the awareness of consumers about the product, so the mode of stepwise advertising (with the same total budget!) can be more profitable from this point of view.

Example 3.

Investigate the dynamics of awareness \(A\left( t \right)\) for the case of Linear Advertising Function \(q\left( t \right).\) Use the same data as in Examples \(1,2.\)

Solution.

In this problem we make our marketing model a bit more complicated. We will assume that the advertising budget is spent over year according to the linear law:

\[q\left( t \right) = {q_0} + \alpha t.\]

The function \(q\left( t \right)\) may be as increasing or decreasing (Figure \(5\)).

Linear advertising functions
Figure 5.
3 limiting cases of linear advertising
Figure 6.

In any case, the total yearly advertising budget remains unchanged (let it be equal to \(U\)). Graphically, this means that the area of all trapezoids (or triangles in the limiting case) shown in Figure \(5\) is the same.

The parameters \({q_0}\) and \(\alpha\) are related by the equation:

\[12\left( {{q_0} + 6\alpha } \right) = U.\]

The left side of this formula is the area of the trapezoid. Then the coefficient \(\alpha\) is expressed in terms of \({q_0}\) as follows:

\[\alpha = \frac{1}{{72}}\left( {U - 12{q_0}} \right).\]

The time dependence of advertising costs will be given by the formula:

\[q\left( t \right) = {q_0} + \frac{{\left( {U - 12{q_0}} \right)t}}{{72}}.\]

Substitute this expression into the general solution \(A\left( t \right)\) and then integrate it (The integral \({\int {{e^{kt}}tdt} }\) can be found using integration by parts). As a result, we obtain the following expression for \(A\left( t \right):\)

\[A\left( t \right) = \frac{b}{k} \left[ {{q_0} + \frac{{U - 12{q_0}}}{{72}}\left( {t - \frac{1}{k}} \right)} \right] + C{e^{ - kt}}.\]

We determine the constant \(C\) from the initial condition \(A\left( {t = 0} \right) = 0.\) This gives:

\[0 = \frac{b}{k}\left[ {{q_0} - \frac{{U - 12{q_0}}}{{72k}}} \right] + C,\;\; \Rightarrow C = - \frac{b}{k}\left[ {{q_0} - \frac{{U - 12{q_0}}}{{72k}}} \right].\]

By substituting \(C\) into the formula for \(A\left( t \right),\) we get:

\[A\left( t \right) = \frac{b}{k}\left[ {{q_0} - \frac{{U - 12{q_0}}}{{72k}} + \frac{{U - 12{q_0}}}{{72}}t} \right] - \frac{b}{k}\left[ {{q_0} - \frac{{U - 12{q_0}}}{{72k}}} \right]{e^{ - kt}} = \frac{b}{k}\left[ {{q_0} - \frac{{U - 12{q_0}}}{{72k}}} \right] \left( {1 - {e^{ - kt}}} \right) + \frac{{b\left( {U - 12{q_0}} \right)}}{{72k}}t.\]

Finally we substitute the known values: \(k = {\frac{1}{4}},\) \(b = 25,\) \(U = 12,000:\)

\[A\left( t \right) = \frac{{25}}{{\frac{1}{4}}}\left[ {{q_0} - \frac{{12,000 - 12{q_0}}}{{72 \cdot \frac{1}{4}}}} \right] \left( {1 - {e^{ - \frac{t}{4}}}} \right) + \frac{{25\left( {12,000 - 12{q_0}} \right)}}{{72 \cdot \frac{1}{4}}}t = 100\left[ {{q_0} - \frac{2}{3}\left( {1000 - {q_0}} \right)} \right] \left( {1 - {e^{ - \frac{t}{4}}}} \right) + 100 \cdot \frac{{1000 - {q_0}}}{6}t.\]

If we set \({q_0} = 1000\) (the mode of homogeneous advertising considered in Example \(1\)), we get the formula found above:

\[A\left( t \right) = 100 \cdot 1000\left( {1 - {e^{ - \frac{t}{4}}}} \right) + 0 = 100,000\left( {1 - {e^{ - \frac{t}{4}}}} \right).\]

Using the general solution \(A\left( t \right),\) we compare the dynamics of awareness for the following limiting cases (see Figure \(6\) above).

The results of the computations are presented in Figure \(7.\)

Awareness of consumers for different linear advertising modes
Figure 7.
Comparison of three advertising scenarios
Figure 8.

The mean values of \(A\left( t \right)\) for the year for the indicated scenarios are also given in the table (Figure \(8\)). As it can be seen, the most aggressive scenario \(2\) can lead to sales growth, though scenario \(1\) has advantages in terms of long-lasting effect. Of course, these conclusions are limited by the accuracy of the model. The dynamics of \(A\left( t \right)\) in a real business environment can be more complex.

Page 1 Page 2