Equations with Derivatives

Calculus 2 · Axiom Academy

When an equation involves a derivative, you don't solve for a number — you solve for a whole function. Watch a population obey one and see why. An equation whose unknown is a function Every equation you have solved so far asked for a number : find the x that makes it true. A differential equation asks something stranger — it ties together a quantity and its own rate of change, and the thing that makes it true is an entire function . You don't have to take that on faith. Watch one play out. Press play. The sweep line marches forward in time; at every instant the curve climbs by exactly k times its current height — so the steeper-and-steeper tangent is the equation acting. What the sweep leaves behind is the whole solution function P(t). The solution isn't a single value — it's a function that gives the population at every moment in time. One equation, a different function for every k The structure never changes — but the single number k decides everything. Drag it and watch the same equation swing from a population that dies out, to one that holds steady, to one that explodes. Each setting is a genuinely different solution function. When k > 0 the function grows; when k < 0 it decays; when k = 0 it stays put. One equation structure, infinitely many solution functions. The starting point picks one curve from a family

This is the written version of the interactive lesson above. See the full Calculus 2 course.