Growth and Decay Problems

Differential Equations · Axiom Academy

A bank account compounding and a radioactive isotope decaying look nothing alike — but they solve the exact same differential equation with the sign of one constant flipped. Two stories, one equation: the rate of change is proportional to the amount that's already there. Drive both stories yourself, then watch where that shared equation actually comes from. Set an interest rate, hit Grow, and watch a deposit compound continuously — the bigger the balance gets, the faster it earns. That's : the growth RATE is proportional to the balance that's already there. An isotope that fades on its own schedule Same shape, opposite sign. Pick a half-life, then drag the clock forward and watch a sample of atoms decay — the amount LEFT determines how fast it keeps decaying. That's : half the atoms disappear every half-life, no matter how many are left. Where y_0 e^ kt actually comes from Beats 1 and 2 aren't two different formulas — they're the SAME separable differential equation, solved once, with k positive for growth and negative for decay. Step through the algebra yourself. One separable equation, , covers both stories: k>0 gives compound growth, k<0 gives radioactive decay, and both solve to the same y(t)=y_0e^ kt . The two "how long" questions are mirror images of each other — doubling time for growth, half-life for decay. Growth — compound interest, unchecked population, viral spread Decay — radioactivity, drug clearance, discharging capacitors

This is the written version of the interactive lesson above. See the full Differential Equations course.