IA ideas library
Differential equation IA ideas for HL (and ambitious SL)
Differential equations describe how something changes, which makes them perfect for epidemics, populations, cooling, falling objects and chemical reactions. They are in both HL courses: AA HL solves them analytically (separation of variables, integrating factors) and AI HL builds coupled systems, slope fields and Euler's method.
A strong DE exploration sets up the equation from assumptions you state, solves or simulates it, compares the result with data, and reflects on which assumption is doing the most damage when the model and reality disagree.
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How differential equations explorations reach the top bands
Personal engagement (C)
State every assumption you build into the equation and why you believe it; collect or choose data that can test those assumptions. Changing an assumption because your data forced you to is strong engagement.
Reflection (D)
Compare the solution with data and identify which assumption does the most damage. Reflect on parameter values: are they physically sensible?
Use of mathematics (E)
SL: At SL, stay with exponential or logistic models fitted to data and explain the rate idea behind them; a DE can be described and its solution used if you explain it.
HL: Set up the DE from assumptions, solve it analytically (separation of variables, integrating factor) or numerically (Euler's method with a step-size study), and compare with data; for AI HL, coupled systems and slope fields are natural.
An HL version
Compare an analytic solution with Euler's method at several step sizes, study how the error changes, or extend to a coupled system.
Differential equations & dynamics ideas (10)
- How long does a skydiver take to reach terminal velocity?AA HLAI HLAmbitious
- Linear or quadratic drag? Testing air resistance with falling coffee filtersAA HLAI HLSolid
- How much of cycling effort goes on air resistance? A coast-down testAA HLAI HLAmbitious
- How fast would flu spread through my school? An SIR modelAI HLAA HLAmbitious
- How long would a lake take to flush out a pollutant?AA HLAI HLSolid
- Can the Lotka–Volterra model explain the lynx and hare cycles?AI HLAA HLAmbitious
- Can mathematics predict a battle? Lanchester's laws and Iwo JimaAI HLAA HLAmbitious
- How does bread dough rise? A logistic model in the kitchenAA SLAA HLAI SLAI HLAccessible
- How fast does news spread through a year group?AI SLAI HLAA SLAA HLSolid
- Does caffeine build up if you drink coffee every few hours?AA SLAA HLAI SLAI HLSolid
Also relevant (10)
- How long does a bottle take to drain? Testing Torricelli's lawAA SLAA HLAI SLAI HLSolid
- Exponential or logistic? Modelling my country's populationAA SLAA HLAI SLAI HLSolid
- Which mug keeps tea hot longest? Newton's law of cooling, comparedAA SLAA HLAI SLAI HLAccessible
- Radioactive decay with dice: does chance produce an exponential curve?AA SLAI SLAA HLAI HLAccessible
- How much vaccination is enough to stop measles?AI SLAI HLAA SLSolid
- Modelling bacterial growth in the biology labAA SLAA HLAI SLAI HLSolid
- How fast did COVID-19 spread at the start? Comparing doubling timesAA SLAA HLAI SLAI HLSolid
- How small must the step be? Euler's method against an exact solutionAA HLAI HLSolid
- From steady to chaotic: the logistic map's period doublingAA HLAI HLAmbitious
- How quickly does a swing die down? Exponential or linear decay?AA SLAA HLAI HLAI SLSolid
Frequently asked questions
Can an SL student write about differential equations?
Yes, if you explain the new mathematics carefully and keep it at a level you fully understand. Euler's method on a spreadsheet is a good route. Criterion E rewards mathematics that is relevant and correct, not the hardest mathematics possible.
What data can I compare a differential equation model with?
Your own cooling or draining measurements, public epidemic data (for example the archived Johns Hopkins COVID-19 repository), population series from the UN or Our World in Data, or motion data from a video tracked in Tracker.
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