 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/14b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Write

E_{n} (t) = P (E, n)

and

C_{n} (t) = P (ES, n)

. The master equation balances the transitions

E_{n} \to C_{n}

at rate

k_{1}

C_{n} \to E_{n}

at rate

k_{1}^{'}

, and

C_{n} \to E_{n + 1}

at rate

k_{2}

\dot{E}_{n} = - k_{1} E_{n} + k_{1}^{'} C_{n} + k_{2} C_{n - 1}, \dot{C}_{n} = k_{1} E_{n} - (k_{1}^{'} + k_{2}) C_{n},

for

n \geq 0

, with

C_{- 1} = 0

 New to topics? Read the docs here!