 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/d/d4/1/b/solution

From (a), combine the three derivative integrals:

t J_{0}^{''} + J_{0}^{'} + t J_{0} = \frac{2}{π} \int_{0}^{π /2} (t sin^{2} x cos (t cos x) - cos x sin (t cos x)) d x .

- \frac{d}{d x} [sin x sin (t cos x)]

. This bracket vanishes at both endpoints, so the integral is zero. Hence

t J_{0}^{''} + J_{0}^{'} + t J_{0} = 0

, also at

t = 0

by the same formula.

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