Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 30K / a

Solution

[t, t + Δ t]

gives

F (x, t) = in f_{u} {c_{t} u^{2} Δ t + F (x + u Δ t, t + Δ t)} + o (Δ t) .

F (x + u Δ t, t + Δ t) = F (x, t) + F_{x} u Δ t + F_{t} Δ t + o (Δ t) .

After subtracting

F (x, t)

, dividing by

Δ t

F_{t} = - u in f {c_{t} u^{2} + F_{x} u}, F (x, h) = x^{2} .

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