Match each real-world statement to the differential equation that models it.

Population grows in proportion to its size
Coffee cools toward room temperature $T_r$
A falling body approaches terminal velocity
Growth slows as capacity $M$ is approached

$y' = ky$
$y' = -k(y - T_r)$
$y' = ky(1 - y/M)$
$v' = g - cv$

Hints

Translate word by word: "in proportion to its size" literally means the rate equals a constant times $y$ .
Whenever something approaches a limit, look for a rate that shrinks to zero exactly at that limit.

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Population grows in proportion to its size → $y' = ky$
Coffee cools toward room temperature $T_r$ → $y' = -k(y - T_r)$
A falling body approaches terminal velocity → $v' = g - cv$
Growth slows as capacity $M$ is approached → $y' = ky(1 - y/M)$

Why

Modelling is translation: each phrase about rates becomes a term in $y'$ . Notice how "approaches" always produces a stabilizing term that vanishes at equilibrium, the mathematical signature of a system settling down.

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Match each real-world statement to the differential equation that models it.

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