Exponential Growth & Decay

The differential equation: dy/dt = ky

Solution: y = y₀eᵏᵗ

k > 0: growth. k < 0: decay.

Example — Population: A population of 1000 grows at 5% per year.

dy/dt = 0.05y. Solution: y = 1000e^(0.05t).

Example — Radioactive decay: Half-life of 10 years.

y = y₀e^(kt). At t=10, y = y₀/2.

y₀/2 = y₀e^(10k). e^(10k) = 1/2. k = ln(1/2)/10 = -0.0693.

Newton's Cooling: dT/dt = k(T - Tₛ) where Tₛ is surrounding temperature. Same separation of variables technique.

AP exam: Expect one question involving dy/dt = ky. Know the solution by heart and how to find k from given conditions.

Reference:

Wikipedia: Exponential Growth

https://en.wikipedia.org/wiki/Exponential_growth