Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 9 / a

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- Solution a

Solution

For real coordinates, work on

y > 0

. The chain rule gives

ξ_{y} = y^{- 1/2}

and

ξ_{yy} = - \frac{1}{2} y^{- 3/2}

. Therefore

\frac{\partial ^{2} u}{\partial y ^{2}} = u^{''} (ξ) (ξ_{y})^{2} + u^{'} (ξ) ξ_{yy} = \frac{u ^{''} ( ξ )}{y} - \frac{u ^{'} ( ξ )}{2 y ^{3/2}} .

The derivatives on the right are with respect to

ξ

. Behaviour at

y = 0

requires separate regularity of the composed function.

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