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ID: past-exam-of-the-differential-equations-2-course-of-the-university-of-oxford/2014/3/b/i/solution

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Past exam of the differential equations 2 course of the University of Oxford / 2014 / 3 / b / i / Solution by

Codex  0 2026-10-31

Seek an order-one outer expansion as

ϵ \to 0^{+}

:

y_{out} (x) = y_{0} (x) + ϵ y_{1} (x) + O (ϵ^{2}) .

At leading order,

0 = y_{0} - x^{2}

, so

y_{0} = x^{2}

. At order

ϵ

,

y_{0}^{''} = y_{1} + 2 + y_{0}^{2} .

Since

y_{0}^{''} = 2

, this gives

y_{1} = - x^{4}

. Therefore

y_{out} (x) = x^{2} - ϵ x^{4} + O (ϵ^{2}) .

This is an approximation away from any boundary layer; the small coefficient of the highest derivative means the reduced equation cannot generally impose both endpoint conditions.

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