Speak Maths · University

Differential equations: a law for how things change

An ordinary equation asks for a number. A differential equation asks for a whole function, and tells you only how fast it changes. Here is how to read one, and how a cup of tea obeys it.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$y^{'} = k y$ Say: y prime equals k yMeans: the rate of change of y is k times y itself: the more there is, the faster it changes
$\frac{d y}{d t} = - k (y - T)$ Say: dy by dt equals minus k times the bracket y minus TMeans: y moves towards T at a rate proportional to its distance from T
$y (0) = y_{0}$ Say: y of zero equals y noughtMeans: the initial condition: the value of y at the start, when t is zero
$y = y_{0} e^{k t}$ Say: y equals y nought times e to the k tMeans: the one solution of y prime equals k y that starts at y nought
$lim_{t \to \infty} y (t) = T$ Say: the limit of y of t, as t goes to infinity, is TMeans: where y settles in the long run: the equilibrium

The story: The cooling cup of tea

Subtitles:

Tap a line to see its words.

$y (t)$ Let y of t be the temperature of a cup of tea, t minutes after it was poured.
$y (0) = 90, T = 20$ The tea starts at 90 degrees. The room stays at 20 degrees.
$\frac{d y}{d t} = - 0.1 (y - 20)$ The tea loses heat at a rate proportional to how much hotter it is than the room. Here the constant is 0.1 per minute.
$y^{'} (0) = - 0.1 (90 - 20) = - 7$ At the very start the tea is falling by 7 degrees every minute.
$u = y - 20, u^{'} = - 0.1 u, u (0) = 70$ Look only at the excess over the room, u. Its rate is minus 0.1 times itself, and at the start it is 70.
$u = 70 e^{- 0.1 t}$ A quantity whose rate is a constant times itself is exponential: u is 70 times e to the minus 0.1 t.
$y = 20 + 70 e^{- 0.1 t}$ Add the room back, and the whole cup is described.
$y (10) = 20 + 70 e^{- 1} \approx 45.8$ After ten minutes the tea is at about 45.8 degrees.
$t \to \infty lim y (t) = 20$ In the long run the tea ends exactly at room temperature, and never below it.

In words this story is 707 letters long; in symbols, 128.

Read it

What does it say?

$y^{'} = 3 y$

$\frac{d y}{d t} = - 0.2 (y - 15)$

$y (0) = 5$

$y = 5 e^{- 2 t}$

$y^{'} = - 0.5 y$

$y (0) = 100, y = 100 e^{0.2 t}$

$t \to \infty lim y (t) = 0$

$\frac{d y}{d t} = 2 (y - 10)$

$y^{'} (0) = - 7$

$y = 20 + 70 e^{- 0.1 t}$

$y^{'} = k y, k < 0$

$u = y - T \Rightarrow u^{'} = - k u$

Write it

Which notation says this?

The rate of change of y is a constant k times y.

The rate of change of y is minus k times the gap between y and T, where k is a positive constant.

The solution of y prime equals k y that starts at y nought when t is zero.

y starts at 100.

y shrinks at a rate proportional to itself, with constant 0.3.

In the long run the temperature settles at 20.

The solution of y prime equals minus 2 y that starts at 7.

A warm bath cools towards the room temperature 18, with constant 0.05.

The number of bacteria in a dish grows at a rate 0.04 times the number present.

At the start the quantity is y nought, and at time t it is y nought times e to the minus 3t.

In the long run the gap between y and T vanishes.

At the start the tea cools at minus k times the gap between its starting temperature and the room’s.

Build it

Tap the tiles in order. One tile does not belong.

The amount at time t is the starting amount times e to the k t.

The rate of change of y is minus k times the gap between y and T.

y prime equals k times y.

y at time zero equals y nought.

The limit of y of t, as t goes to infinity, is T.

The rate of change of y is minus 0.1 times the gap between y and 20, and y starts at 90.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 Which curve obeys which law?

Three curves, four facts. Judge each sentence about them: true or false?

	$y^{'} = 2 y$	$y^{'} = - 2 y$	$y (0) = 1$	$lim_{t \to \infty} y (t) = 0$
$e^{2 t}$	✓		✓
$3 e^{2 t}$	✓
$e^{- 2 t}$		✓	✓	✓

A tick means the curve in that row has the fact in that column.

$y = e^{2 t} \Rightarrow (y^{'} = 2 y \land y (0) = 1)$
$y = 3 e^{2 t} \Rightarrow y (0) = 1$
$y = e^{- 2 t} \Rightarrow y^{'} = 2 y$
$y = e^{- 2 t} \Rightarrow lim_{t \to \infty} y (t) = 0$
$y = 3 e^{2 t} \Rightarrow y^{'} = 2 y$
$y = 3 e^{2 t} \Rightarrow lim_{t \to \infty} y (t) = 0$

2 Laws and their stories

Pair each line with what it says.

Tap a line, then its meaning.

3 The cooling tea

A cup of tea cools towards the room’s temperature T, at a rate proportional to the gap y − T. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 Where does it end?

After a very long time the tea is at room temperature T. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

5 A clock on the decay

Decode the line: what number is y at the time t = 2 ln 2?

$y^{'} = - \frac{1}{2} y, y (0) = 8, y (2 ln 2) = ?$

6 The long run

Where does y settle in the end?

$\frac{d y}{d t} = - 3 (y - 12), y (0) = 100, t \to \infty lim y (t) = ?$

7 Write the law of cooling

The hotter a cup of tea is compared with the room, the faster it cools. Write that as a law, with y the tea’s temperature, T the room’s and k > 0. Then compare.

Your version

8 Start with a hundred

A colony of cells grows at a rate proportional to its size, and at the start there are 100 cells. Write the law with its starting condition, or write the solution itself.

Your version

9 Four cups

Three hot drinks and one cold one sit in a room at 20. Each row is a cup’s temperature over time. Judge each sentence.

	$y (0) = 80$	$lim_{t \to \infty} y (t) = 20$	$y^{'} (0) < 0$	$y^{'} = - 0.1 (y - 20)$
$y_{1} = 20 + 60 e^{- 0.1 t}$	✓	✓	✓	✓
$y_{2} = 20 + 60 e^{- 0.3 t}$	✓	✓	✓
$y_{3} = 25 + 60 e^{- 0.1 t}$			✓
$y_{4} = 20 - 15 e^{- 0.1 t}$		✓		✓

A tick means the cup in that row has the fact in that column.

$y_{1} (0) = y_{2} (0)$
$lim_{t \to \infty} y_{3} (t) = 20$
$y_{4}^{'} (0) > 0$
$y_{4}^{'} = - 0.1 (y_{4} - 20)$
$y_{2}^{'} = - 0.1 (y_{2} - 20)$
$y_{3} (0) = 80$

10 Four rates

Pair each law with the story it tells.

Tap a line, then its meaning.

11 Three threes and a fourth

The same number 3 says four different things. Pair each line with its meaning.

Tap a line, then its meaning.

12 The wrong direction

The solution of y′ = 2y that starts at 5 grows: it is 5 times e to the 2t. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

13 Where it begins

The tea is poured at 90 degrees: at time zero, y is 90. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 Doubling in one

Something grows by its own law, starts at 10 and is 20 one unit of time later. What is k?

$y^{'} = k y, y (0) = 10, y (1) = 20 ∴ k = ?$

15 Halfway to the room

The tea started at 90 in a room at 20. When is it at 55, halfway down?

$y = 20 + 70 e^{- 0.1 t}, y (t) = 55 ∴ t = ?$

16 The long fade

y decays with constant 3 from a start of 5. Where does it end?

$y^{'} = - 3 y, y (0) = 5 ∴ t \to \infty lim y (t) = ?$

17 A bath goes cold

A bath at 40 degrees cools in a room at 20; the hotter it is, the faster it cools, with constant k. Write the law with its starting value, or the solution itself. Then compare.

Your version

18 The room wins

However hot the tea was, in the long run its temperature is the room’s, T. Write that. Then compare.

Your version

Tell your own

Write a two-line story of something that fades or grows by itself — a warm bath cooling, a rumour spreading — in notation only: a rate law, a starting value, a solution.

Kept on this device only.

Differential equations: a law for how things change

The alphabet

The story: The cooling cup of tea

Read it

Write it

Build it

Challenges

1 Which curve obeys which law?

2 Laws and their stories

3 The cooling tea

4 Where does it end?

5 A clock on the decay

6 The long run

7 Write the law of cooling

8 Start with a hundred

9 Four cups

10 Four rates

11 Three threes and a fourth

12 The wrong direction

13 Where it begins

14 Doubling in one

15 Halfway to the room

16 The long fade

17 A bath goes cold

18 The room wins

Tell your own

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