Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 3 / d

Solution

f (x) = x

gives the exact moment equation

\frac{d m}{d t} = ⟨ \frac{α _{1} α _{3}^{2} V ^{3}}{α _{4}^{2} X ^{2}} - \frac{α _{2} X ( X - 1 )}{V} ⟩ .

Under the stated moment closure this becomes the same expression evaluated at

X = m

m = V \overset{x}{ˉ}

and let

V \to \infty

to obtain

\frac{d x ˉ}{d t} = \frac{α _{1} α _{3}^{2}}{α _{4}^{2} x ˉ ^{2}} - α_{2} \overset{x}{ˉ}^{2} .

Its positive stable equilibrium is

t \to \infty lim \overset{x}{ˉ} (t) = (\frac{α _{1} α _{3}^{2}}{α _{2} α _{4}^{2}})^{1/4} .

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