 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 336

Now
$$
f=-u^2u_t
=R^3\cos^2\theta\sin\theta.
$$
Therefore
$$
R'=-R^3
\langle\cos^2\theta\sin^2\theta\rangle
=-\frac{R^3}{8},
$$
whereas $R\phi'=0$ because $\langle\cos^3\theta\sin\theta\rangle=0$. Integrating with $R(0)=1$ gives
$$
\frac d{dT}R^{-2}=\frac14,
$$
and hence
$$
\boxed{
R(T)=\left(1+\frac T4\right)^{-1/2},
\qquad
\phi(T)=0}.
$$
The leading uniformly slow-decaying oscillation is
$$
\boxed{u(t)\sim
\left(1+\frac{\epsilon t}{4}\right)^{-1/2}
\cos t.}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / ii / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / ii / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)