Applications Summary

Differential Equations · Axiom Academy

SUMMARY Applications of First-Order Equations A recap of how one differential equation — a rate proportional to state — models exponential growth and decay, Newton's law of cooling, and population growth with limited resources. Three models, one shared skeleton: rate of change depends on current state — exponential ( dy/dt=ky ), cooling ( dT/dt=-k(T-T_a) ), and logistic ( dy/dt=ry(1-y/K) ) are all separable first-order equations. Unconstrained exponential growth is a modeling idealization — real populations, drug concentrations, and temperatures eventually hit a limit or an equilibrium. The logistic equation fixes exponential growth's blind spot by adding a carrying capacity K ; population growth slows as it approaches K instead of blowing up forever. Newton's law of cooling/heating is exponential decay toward an ambient value, not toward zero — every object relaxes to its surroundings' temperature. Harvesting a logistic population (fishing, forestry) has a hard ceiling: the Maximum Sustainable Yield is capped at rK/4 , taken at y=K/2 — harvest past that and the population collapses. Core Concept Exponential Growth & Decay Rate of change proportional to the current amount. k>0 gives runaway growth (bacteria, unconstrained populations); k<0 gives decay toward zero (radioactive isotopes, drug concentration in the bloodstream). C=y(0) is fixed by the initial condition. When to use: "amount changes at a rate proportional to itself," no external limit mentioned.

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