Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 33A / b

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- Solution b

Solution

On the jet space of a scalar ordinary differential equation, the

n

th prolongation of a vector field is

pr^{(n)} V = V + \sum_{j = 1}^{n} η^{(j)} \partial_{u^{(j)}}, η^{(j)} = D_{x} η^{(j - 1)} - u^{(j)} D_{x} ξ, η^{(0)} = η,

where

D_{x}

is the total derivative operator. In particular,

pr^{(1)} V = ξ \partial_{x} + η \partial_{u} + [η_{x} + (η_{u} - ξ_{x}) u^{'} - ξ_{u} (u^{'})^{2}] \partial_{u^{'}} .

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