Exponential Models

Differential Equations · Axiom Academy

Deriving y = Ce^ kt from — and seeing how one sign flip splits every application into growth or decay. 1. Deriving the General Solution Start with the differential equation where the rate of change is proportional to the current value — k is the proportionality constant and y(t) is the quantity at time t : This equation separates : put every y on the left, every t on the right, then integrate both sides. 2. The Sign of k Determines Everything The constant k is the growth/decay rate, and its sign alone completely changes the solution's behavior. Watch the curve y=Ce^ kt as k sweeps continuously from negative to positive: The quantity increases without bound. Every instant, you add a fixed percentage of whatever you currently have — so the more you have, the faster it grows. The quantity shrinks toward zero. Every instant, you lose a fixed percentage of what remains — so the closer to zero, the slower it falls, never quite reaching it. The same shape — one curve, one constant k — reappears throughout science and everyday life under different names and different variables: Finance: compound interest P(t)=P_0e^ rt , where r is the interest rate — money grows exponentially once interest itself starts earning interest. Biology: population growth N(t)=N_0e^ kt when resources are unlimited — colonies double at regular intervals. Physics: radioactive decay , where is the decay constant — the half-life is when N=N_0/2 .

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