The unit where calculus starts describing the real world — growth, decay, motion, mixing. Five interactive tools that turn slope fields, Euler’s method, and separable equations from procedures you memorize into things you can watch.
Work them in order for a full walkthrough of the unit, or jump straight to whatever's giving you trouble.
Before you solve one, see what a differential equation actually says: a rule tying a quantity to its own rate of change. Model a real situation as a DE, then learn to verify when a function is a genuine solution.
The picture that makes differential equations click. At every point a tiny segment shows the slope a solution must have there — drop in a starting point and watch the curve thread its way through the field.
When a differential equation can’t be solved exactly, you can still march forward. Step along the tangent line, recompute the slope, step again — and watch the approximation tighten as the step size shrinks.
The main technique for solving by hand. Get every y on one side and every x on the other, integrate both sides, then use an initial condition to pin down the single curve you actually want.
Two models that describe half the natural world. Exponential growth that never lets up, versus logistic growth that levels off into its carrying capacity — watch the S-curve bend and flatten against the ceiling.
Put it together the way the real BC exam does — mixed multiple choice first, then a full differential-equations free-response.
You pick the approach: mixed multiple choice across all nine topics with worked explanations and a weak-spot finder, plus a chained free-response in the classic BC differential-equations mold — slope field, an Euler step, separation, and a growth model.
Begin practicing →From what a differential equation is, through slope fields, Euler’s method, and separation of variables, to exponential and logistic growth — and a capstone that mixes it all together the way the real exam does. Everything verified, nothing hand-wavy.