Differential equations in context

Forming the model

Turn the words into an equation with a constant k. Newton's law of cooling: the rate of cooling is proportional to the difference from room temperature, so dθ/dt = -k(θ - 20).

Finding constants

Solving gives a constant of integration (from the initial value) plus k. Use the first condition (often t = 0) for the constant, then a second condition for k.

Interpreting

• Long term: what happens as t gets large?
• Does the model ever reach a value (for example, does a tank empty)?
• Limitations: dP/dt = kP predicts unlimited growth, which is unrealistic.

Worked example

dθ/dt = -k(θ - 20), θ = 90 when t = 0 and θ = 55 when t = 10. Find k.

1. ∫ 1/(θ - 20) dθ = -∫ k dt gives θ - 20 = Ae^-kt
2. t = 0: A = 70
3. t = 10: 35 = 70e^-10k, so e^-10k = 1/2

Answer: k = ln 2/10 ≈ 0.0693

Check yourself

1. For θ = 20 + 70e^-kt with k = ln 2/10, find θ when t = 20.
   Show answer

   37.5. e^-20k = (1/2)² = 1/4, so θ = 20 + 17.5 = 37.5.

2. For θ = 20 + 70e^-kt with k = ln 2/10, find the time when θ = 30, to 1 decimal place.
   Show answer

   28.1. 70e^-kt = 10, so e^-kt = 1/7 and t = ln 7/k = 28.1.

3. For θ = 20 + 70e^-kt with k > 0, what happens to θ in the long term?
   • θ approaches 90
   • θ approaches 0
   • θ approaches 20
   • θ increases without limit
   Show answer

   θ approaches 20. e^-kt tends to 0, so θ tends to the room temperature 20.

4. dP/dt = kP, P = 1000 when t = 0 and P = 1500 when t = 2. Find P when t = 4.
   Show answer

   2250. P = 1000e^kt with e^2k = 1.5. At t = 4: e^4k = 1.5² = 2.25, so P = 2250.

5. Water leaks from a tank so that the depth h decreases at a rate proportional to the square root of h. Which equation models this (k > 0)?
   • dh/dt = k sqrt(h)
   • dh/dt = -kh²
   • h = -k sqrt(t)
   • dh/dt = -k sqrt(h)
   Show answer

   dh/dt = -k sqrt(h). Decreasing: negative. Proportional to sqrt(h): k sqrt(h).

6. dh/dt = -0.4 sqrt(h), with h = 9 when t = 0. Find the time when the tank is empty.
   Show answer

   15. ∫ h^-1/2 dh = -∫ 0.4 dt gives 2sqrt(h) = -0.4t + c. t = 0: c = 6. Empty when h = 0: 0.4t = 6, so t = 15.

7. What is the main limitation of the model dP/dt = kP (k > 0) for a population?
   • It predicts the population falls to zero
   • It has no constant of integration
   • It only works when P is negative
   • It predicts unlimited growth
   Show answer

   It predicts unlimited growth. Its solution P = Ae^kt grows for ever, but resources are limited.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.