Determining the average rate of change of a product's price over time

Question

I'm currently studying a model that describes the cost of 1 kg of a product $p$ in dollars over time, given by the equation $$\frac{d^3p}{dt^3} + a\frac{dp}{dt} = b,$$ where $a = 9[year^{ - 2}], b = 27[dollar \cdot year^{ - 3}]$.

My task is to find the average rate of price change of $p$ over the time interval $[0; + \infty)$ dollars per year, which is represented by $\underset{t\rightarrow +\infty}{\mathrm{\lim}}\frac{p(t)}{t}$.

I've attempted to use the given model to analyze the rate of change of $p(t)$ over time but haven't been able to determine the method to calculate the average rate of price change over an infinite time interval.

Could someone provide guidance or steps to compute the average rate of change of $p$ with respect to time over the time interval $[0; + \infty)$ in dollars per year, considering the given model equation?

Answer 1

We first solve the differential equation:

$$ \frac{d^3p}{dt^3} + 9 \frac{dp}{dt} = 27 $$

The characteristic equation is:

$$ \lambda^3 + 9 \lambda = 0 $$ $$ \lambda(\lambda^2 + 9) = 0 $$

Which means $\lambda = 0$ and:

$$ \lambda = \pm \sqrt{-9} = \pm 3i $$

The characteristic solution to the equation $p_c$ is:

$$ p_c = c_1e^{3ti} + c_2 e^{-3ti} + c_3 $$

Now, we take the particular solution to the equation $p_p$ to be:

$$ p_p = ct + d $$

Substituting into the differential equation we get $ 9c = 27 $ and so $c = 3$. This means the general solution of the differential equation is:

$$ p = p_c + p_p = c_1e^{3ti} + c_2 e^{-3ti} + 3t + c_3 $$

which gives the price of the product at time $t$. The average change in price $A$ over a period $T$ is:

$$ A = \frac{p(T) - p(0)}{T} $$

$$ = \frac{c_1e^{3Ti} + c_2 e^{-3Ti} + 3T + c_3 - c_1 - c_2 - c_3}{T} $$ $$ = \frac{c_1e^{3Ti} + c_2 e^{-3Ti} + 3T + c_3}{T} $$ $$ = \frac{c_1 \cos(3T) + c_2 \sin(3T)i + 3T + c_3}{T} $$

The limit as $T$ grows large gives us:

$$ A = \lim_{T \rightarrow \infty} \frac{c_1 \cos(3T) + c_2 \sin(3T)i + 3T + c_3}{T} $$ $$ A = 3 $$

So the long time average price increase $A$ is $3$ dollars per year.