Differential Equations in Nature

Differential Equations · Axiom Academy

Differential Equations in Nature One equation, three phenomena — bacteria multiplying, a medicine dose fading, a coffee cooling. Different stories, the same rate law. Look closely and the SAME shape shows up everywhere: a quantity's rate of change depends on the quantity itself. Meet it three times, in three different disguises. Set a growth rate, hit Grow, and watch a bacteria colony multiply — the more bacteria there already are, the faster new ones appear. That's : the growth RATE is proportional to the population that's already there. A dose that fades on its own schedule Same shape, opposite sign. Drag the clock forward and watch how much of a medication dose is still in the bloodstream — the amount LEFT determines how fast it clears. That's : decaying twice as much when there's twice as much left. A cup of coffee settling toward the room A third disguise: the rate depends on a DIFFERENCE, not the raw amount. Drag time forward and watch a hot cup of coffee cool toward room temperature — fast at first, slower as it gets close. That's Newton's Law of Cooling, . Three phenomena, one shape: a quantity whose rate of change is tied to the quantity itself. Growth ( k>0 ) and decay ( k<0 ) are the SAME equation with the sign flipped; cooling is the same idea one layer deeper — the rate depends on a difference , not the raw amount. Every equation you meet in this course is one of these families, or a combination of them. Population growth — bacteria, colonies, compound interest

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