Match each differential equation to the story it tells.

$y' = ky,\; k > 0$
$y' = -ky,\; k > 0$
$y' = k(A - y)$
$y' = ky(1 - y/M)$

Unchecked growth (compound interest)
Growth that saturates (logistic population)
Decay toward zero (radioactivity)
Approach to a ceiling (coffee cooling)

Hints

Ask of each equation: where is $y'$ zero? That is where the system can rest.
The logistic starts exponential (small $y$ ) and strangles itself as $y \to M$ .

Show the answer

$y' = ky,\; k > 0$ → Unchecked growth (compound interest)
$y' = -ky,\; k > 0$ → Decay toward zero (radioactivity)
$y' = k(A - y)$ → Approach to a ceiling (coffee cooling)
$y' = ky(1 - y/M)$ → Growth that saturates (logistic population)

Why

Four rate laws cover a remarkable share of science. Reading equilibria ( $y'=0$ ) and signs of $y'$ gives you the full qualitative story before any solving, often that is all you need.

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Match each differential equation to the story it tells.

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