How do differential equations model things like population growth or radioactive decay?
Both are cases where the rate of change is proportional to the current amount: dy/dt = ky.
Both are cases where the rate of change is proportional to the current amount: dy/dt = ky. For a population with plentiful resources k > 0, and solving gives exponential growth y = y₀e^{kt}. For radioactive atoms k < 0, giving exponential decay to a half-life. The differential equation captures the rule ('change is proportional to how much you have'), and integrating it turns that rule into a formula predicting the amount at any time.
Key point
dy/dt = ky models proportional change; its solution y = y₀e^{kt} gives growth (k>0) or decay (k<0).
Common mistake
assuming growth/decay is linear rather than proportional to the current amount.
Memory tip
'rate ∝ amount' → dy/dt = ky → exponential behaviour.
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