 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

2021

Paper 344

For a planar equilibrium, multiply $\mu=0$ by $\phi'$ and use the <chain rule>. The result is
$$
\frac d{dx}\left[
f(\phi)-\frac{\kappa(\phi)}2(\phi')^2
\right]=0.
$$
Matching to either bulk phase fixes the constant:
$$
\boxed{
f(\phi)-\frac{\kappa(\phi)}2(\phi')^2=f(\phi_B)
}.
$$
Consequently
$$
\frac{dx}{d\phi}
=\pm\sqrt{\frac{\kappa(\phi)}
{2[f(\phi)-f(\phi_B)]}},
$$
and the inverse profile is
$$
\boxed{
x(\phi)-x_0
=\pm\int_0^\phi
\sqrt{\frac{\kappa(\psi)}
{2[f(\psi)-f(\phi_B)]}}\,d\psi
}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 1 / e / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 1 / e / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 1 / e / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)