 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/9b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For a comoving region,

E = ρ V

and

V = a^{3} V_{0}

. The work law gives

d (ρ V) = - P d V,

so differentiation with respect to time yields

V \overset{ρ}{˙} + (ρ + P) \dot{V} = 0.

Since

\dot{V} / V = 3 \overset{a}{˙} / a

V

\overset{ρ}{˙} + 3 \frac{a ˙}{a} (ρ + P) = 0 .

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