Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 2 / c / i

Set $r=r_0{x}^2$, $\Sigma ={\Sigma }_0{x}^{-3}S$, and $\bar{\nu }={\bar{\nu }}_0{x}^{2m-3n}{S}^n$. Then ${\partial }_r=(2r_{0}x{)}^{-1}{\partial }_x$. Substitution into the viscous evolution of an accretion disk equation, with $\tau =3{\bar{\nu }}_{0}t/(4r_0^2)$, cancels the common dimensional factors and gives