Topic · full course live
Differential equations
The laws of nature, written as change.
Nature rarely tells you what a quantity is — it tells you how the quantity changes. “Cooling is proportional to temperature difference.” “Growth is proportional to population.” A differential equation is that sentence in symbols; solving it reveals the future.
Your progress
5 lessons
The big ideas
Equations about rates
y′ = ky says: this thing grows in proportion to its size. The unknown isn't a number — it's a whole function, the curve that obeys the law.
Slope fields show the flow
Draw the slope the equation demands at every point, and solutions appear as curves following the grain — you can see the answer before solving anything.
Few solve exactly; all solve numerically
Most real equations have no tidy formula. Computers step them forward in tiny slices — the same Riemann thinking you learned in integrals, running every simulation on earth.
The course — start at lesson one
- 01Reading rate lawsRate-laws in plain English, and slope fields that show every solution.Hard
- 02Exponential growth & decayy′ = ky: the single most important equation.Hard
- 03Predicting step by stepEuler's method — how computers solve what formulas can't.Hard
- 04Systems in motionEpidemics, predator–prey, and orbits: equations in conversation.Hard
- 05Second-order equations & oscillationWhere waves come from: characteristic roots, ringing, resonance.Hard
Out in the world
Epidemics
The SIR model — three linked differential equations — guides real public-health decisions.
Spacecraft & orbits
Missions navigate by integrating Newton's laws forward through time.
Climate & weather
Forecasts are massive systems of differential equations stepped forward on supercomputers.
This topic connects to