7.1 Modeling Situations with Differential Equations

7.1 Modeling Situations with Differential Equations

A differential equation does not usually tell you a function directly; it tells you how the function must change. That makes it a compact model of growth, motion, cooling, population change, or any situation in which a quantity’s rate depends on other quantities.

Investigative question: If you know a quantity’s rate of change, how can you describe the quantity itself—and how can you prove that a proposed function really works?

From words to a differential equation

A differential equation is an equation involving an unknown function and one or more of its derivatives. To model a situation, translate the verbal description into relationships among quantities before attempting to solve anything.

Verbal statement	Mathematical model
The rate of change of $y$ with respect to $t$ is proportional to $y$	$\dfrac{dy}{dt}=ky$
The rate of change of $y$ is proportional to the amount of $y$ and decreases as $y$ approaches a capacity	$\dfrac{dy}{dt}=ky\left(1-\dfrac{y}{M}\right)$
The rate of change of position is velocity	$\dfrac{ds}{dt}=v(t)$

In these models, $k$ is a constant of proportionality, $t$ is often time, and $M$ may represent a limiting capacity. The equation $\dfrac{dy}{dt}=ky$ says that the relative rate of change is constant: a larger amount of $y$ produces a larger absolute change.

Worked modeling example. A population $P$ grows at a rate proportional to its current size. Because “rate of change” means a derivative and “with respect to time” means differentiation by $t$, the model is

$$ \frac{dP}{dt}=kP. $$

If the population decreases rather than grows, then $k<0$. If it grows, then $k>0$. The sign and units matter: if $P$ is measured in organisms and $t$ in hours, then $k$ has units of hours$^{-1}$, so that $kP$ has units of organisms per hour.

Writing the equation is the modeling step. Checking that a proposed function actually satisfies it is the work of Topic 7.2.

AP skills in action

Topic 7.1 develops Skill 2.C: Use appropriate mathematical procedures by translating verbal relationships into differential equations and using antidifferentiation appropriately. Topic 7.2 develops Skill 3.G: Confirm that solutions are accurate and appropriate by differentiating a proposed function and checking the original equation.

Retrieval check: A cup of coffee at temperature $T$ cools at a rate proportional to the difference between $T$ and the room temperature of $20$ °C. Write a differential equation for $T$. One model is $\dfrac{dT}{dt}=k(T-20)$ with $k<0$: the derivative is the rate, $T-20$ is the quantity the rate is proportional to, and the negative constant makes a cup that is hotter than the room cool down.