 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/1/c/ii/solution

(1 - x^{2})^{- 1/2} + (1 - x^{2})^{1/2} = w (2 - x^{2}) = w (\frac{3}{2} y_{0} - \frac{1}{2} y_{2}) .

γ^{2} = 5

, part (i) gives

Z = (\frac{1}{5} + \frac{ε ^{2}}{20}) y_{0} + \frac{ε}{4} y_{1} + \frac{ε ^{2}}{4} y_{2} + O (ε^{3}) .

y_{1}

term and all cross terms vanish. Using

I_{0} = π

and

I_{2} = π /2

, the integral is

\frac{3}{2} π (\frac{1}{5} + \frac{ε ^{2}}{20}) - \frac{1}{2} \frac{π}{2} \frac{ε ^{2}}{4} + O (ε^{3}) = \frac{3 π}{10} + \frac{π}{80} ε^{2} + O (ε^{3}) .

The remainder can be integrated against this integrable weight in the regular expansion, so it retains the stated order.

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