CAIE A-Level Mathematics 3.8.4 Interpreting Differential-Equation Models

CAIE A-Level Mathematics 3.8.4 Interpreting Differential-Equation Models
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise interpreting differential-equation solutions by substituting contextual values and taking limits to identify predicted times, equilibrium levels and long-run behaviour.

How this is tested

  • substitute a target radius, depth or population into the solved model and calculate its time
  • take the relevant exponential term to zero as t tends to infinity to find the limiting value
  • state the contextual meaning of an equilibrium or limiting value rather than only its algebra

Question 10(c)

[Maximum number: 1]

In a chemical reaction, a compound X is formed from two compounds Y and Z.
The masses in grams of X, Y and Z present at time t seconds after the start of the reaction are x, 10-x and 20-x respectively. At any time the rate of formation of X is proportional to the product of the masses of Y and Z present at the time. When t=0, x=0 and dx dt=2\frac{\mathrm{d} x}{\mathrm{~d} t}=2.

State what happens to the value of x when t becomes large.