Interpreting solutions of differential equations in context

9 minA Level3.8.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can explain what the solution of a differential equation means in its real situation: the value the quantity approaches, the range of times the model describes, and the meaning of each constant in the formula.

The formula is only half the answer

Solving a differential equation gives a formula. Interpreting the solution means translating that formula back into the situation it came from.

Three questions come up again and again.

What value does the quantity approach in the long run, and what does that value mean?
Which values of $t$ (or $x$ ) does the model actually describe?
What does each number in the formula stand for?

The answers are sentences about the real situation, not symbols. "The tea cools towards the room temperature of $20\,^\circ\text{C}$ " is an interpretation; " $\theta = 20$ " on its own is not.

Limiting values

Many models settle down. If $k > 0$ , the exponential term $e^{-kt}$ shrinks towards $0$ as $t$ grows.

So any formula of the shape

$\text{quantity} = \text{fixed number} + (\text{something})\,e^{-kt}$

creeps closer and closer to that fixed number.

That fixed number is called the limiting value. The quantity never quite reaches it in finite time; it only approaches it.

Model	Solution	Limiting value	What it means
a cooling body	$\theta = 20 + 70e^{-0.05t}$	$20$	the room temperature
a population in a habitat	$P = \dfrac{1000}{1 + 9e^{-0.2t}}$	$1000$	the largest population the habitat supports
a filling tank	$V = 500(1 - e^{-0.2t})$	$500$	the volume when the tank is full

For the population formula, the reasoning is: as $t \to \infty$ the term $e^{-0.2t} \to 0$ , so the denominator $1 + 9e^{-0.2t} \to 1$ and $P \to \dfrac{1000}{1} = 1000$ .

In an answer, give the limiting value with its units and say what it represents.

Worked example: Reading a cooling model

A cup of tea cools so that $\frac{d\theta}{dt} = -0.05(\theta - 20)$ , where $\theta$ is the temperature in $^\circ\text{C}$ and $t$ is the time in minutes. Initially $\theta = 90$ . The solution is

$\theta = 20 + 70e^{-0.05t}$

(a) What is the temperature after a long time? (b) What does the number 20 represent? (c) Why does the model not apply for $t < 0$ ?

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.