Worlds of differential equations
About this app
An introduction to differential equations: nine running worlds from everyday life, physics, biology, medicine, epidemiology, economics and psychology (coffee, parachute, bacteria, spring, hares and foxes, epidemic, tablets, Solow growth, Romeo and Juliet). Arranged by curve shape, only three shapes remain. In detail: slope arrow = right-hand side of the ODE, draggable initial value, family of solutions, equilibrium.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What a differential equation is
A differential equation (ODE) does not describe where something is, but how fast it is changing right now, depending on its current state. For coffee the rule is T′ = −k(T − T_U): the larger the gap to room temperature, the faster it cools. Together with an initial value the equation fixes the whole course over time. Its solution is therefore not a number but a function. Other initial values give other curves of the same family of solutions.
Nine worlds, three curve shapes
The app shows nine famous models as running scenes: Newton’s law of cooling, a parachute jump with air drag, logistic growth of bacteria, the damped spring, the predator–prey model of Lotka and Volterra, the SIR model of an epidemic, the drug level under regular dosing, the Solow growth model of economics and Strogatz’s Romeo and Juliet. Arranged by curve shape, three shapes remain: equilibrium, growth with a limit and oscillation.
What you can try
Tap a tile to open a world large. On the left the scene runs, on the right the solution curve grows. The slope arrow at the current point is exactly the right-hand side of the ODE. Drag the start point at t = 0 and watch the same rule produce a different curve. The sliders change the cooling constant, the basic reproduction number R₀, the dosing interval or the saving rate. The Equation panel shows the ODE in the scene’s colours, in words, with live values and, where one exists, the solution formula.
Frequently asked questions
- What is a differential equation in simple terms?
- A differential equation is an equation that contains an unknown function and its derivatives. It describes a rule for change: how fast a quantity changes depends on its current value. Together with an initial value it determines the whole course over time.
- Where do differential equations appear in everyday life?
- Wherever something changes over time: coffee cooling down, falling with air drag, the drug level in the blood, bacterial growth, epidemics, oscillations of springs and bridges, predator–prey cycles and growth models in economics.
- Why does a differential equation have many solutions?
- The equation only fixes the rate of change, not the starting point. Every initial value gives its own solution curve; together they form the family of solutions. Only an initial value such as T(0) = 90 °C picks exactly one curve. This is called an initial value problem.
- What can I adjust in the app?
- You choose one of nine worlds, arrange the gallery by field or curve shape, drag the initial value, change two parameters per world, show the family of solutions and scrub time. An explain mode demonstrates the key effects.
Subject: Differential equations
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