IB Maths AI HL
Differential Equations
Paper 1 & 2
~5 min read
Modelling with Differential Equations
A differential equation links a quantity to its rate of change, which makes it the natural tool for modelling things that change over time. The skill here is translation: turning a sentence like “the rate of change is proportional to…” into an equation with a derivative and a constant of proportionality k.
📘 What you need to know
- “Rate of” → derivative: phrases like “rate of change/growth of” signal a derivative, usually with respect to time t.
- Proportional → k: “y is proportional to x” means y = kx for a constant k.
- Combine them: “rate of change of P ∝ P” becomes dPdt = kP.
- Sign for decrease: take k > 0 and use −k when the quantity is falling.
- k is usually unknown: you find it later, from conditions, when solving.
- Read the “function of”: ∝ √A gives k√A; ∝ the difference (T − Ta) gives k(T − Ta).
From words to an equation
The proportionality model
rate of change of P ∝ P ⟶ dPdt = kP (growth) or dPdt = −kP (decay), k > 0
✓ standard modelling set-up for AI HL
🤔 Why use −k instead of just letting k be negative?
Both work mathematically, but keeping k > 0 and writing the sign explicitly makes the model’s behaviour clear at a glance: a leading “+” means the quantity grows, a leading “−” means it decays. For Newton’s cooling, dTdt = −k(T − Ta) with T > Ta guarantees the rate is negative, so the object cools — exactly what you’d expect.
🧠 “Rate equals k times the thing”
Find the rate (a derivative in t), find the thing it’s proportional to (the function in the sentence), and join them with k. Add a minus sign if the quantity is decreasing.
Building the model
🧭 Recipe — set up a differential equation
- Spot the rate: “rate of change of Q” → dQdt.
- Identify the function it’s proportional to (the quantity, its root, a difference, …).
- Write the proportionality with a constant k.
- Fix the sign: k > 0; use −k if the quantity decreases.
- State k as a constant of proportionality (its value comes later).
Decode the phrase: “directly proportional to the square root of A” → k√A; “proportional to the difference between T and Ta” → k(T − Ta); “proportional to the population” → kP.
Worked examples
WE 1In a pond, the rate of change of the area A covered by algae is directly proportional to the square root of that area. Write a differential equation.
Rate → dAdt; proportional to √A.
rate of change of A → dAdt
∝ √A → k√A
dAdt = k√A (k a constant of proportionality)
WE 2Newton’s Law of Cooling: the rate of change of an object’s temperature T is proportional to the difference between T and the ambient temperature Ta. The object starts warmer than its surroundings. Write the equation.
The object cools, so the rate must be negative while T > Ta.
∝ (T − Ta), and T > Ta so (T − Ta) > 0
temperature decreasing → need a minus sign
dTdt = −k(T − Ta), k > 0
−k with (T − Ta) > 0 makes dTdt negative ✓
WE 3A bacteria population P grows at a rate proportional to its current size. Write a differential equation.
Growth proportional to the population itself.
rate of growth of P → dPdt
∝ P, and growing → +k
dPdt = kP, k > 0
WE 4A radioactive sample of mass m decays at a rate proportional to the mass remaining. Write a differential equation.
Decay means the mass is decreasing, so use −k.
rate of change of m → dmdt
∝ m, decreasing → −k
dmdt = −km, k > 0
WE 5A tank’s water height h falls at a rate proportional to the square root of the height (Torricelli’s law). Write a differential equation.
Falling height, proportional to √h.
rate of change of h → dhdt
∝ √h, falling → −k
dhdt = −k√h, k > 0
💡 Top tips
- “Rate of” is the trigger — expect a derivative, usually in t.
- Translate the “function of” literally: square root, difference, product, etc.
- Keep k > 0 and put the sign in front to show growth or decay.
- Check the sign makes sense — does your equation make the quantity move the right way?
- Don’t try to find k yet — it’s a constant of proportionality at this stage.
- Define your variables and state what k represents.
⚠ Common mistakes
- Wrong sign — a “+” for a decreasing quantity (or vice versa).
- Dropping the constant k from the proportionality.
- Misreading the function — using A instead of √A, or T instead of (T − Ta).
- Differentiating with respect to the wrong variable — usually it’s time t.
- Trying to find k before a condition is given.
Next up — Slope Fields. You can now build a differential equation from a context and solve a separable one. But not every equation can be solved with neat algebra — so the next topic shows how to visualise the solutions instead: a slope field draws short tangent lines from dydx at many points, letting you sketch solution curves without ever solving the equation.
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