 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/8/iv/solution

Two integrations give the general reduction of order formula

v (x) = C_{1} + C_{2} \int^{x} \frac{d t}{( 1 - t ^{2} ) P _{n} ( t ) ^{2}}, y (x) = P_{n} (x) [C_{1} + C_{2} \int^{x} \frac{d t}{( 1 - t ^{2} ) P _{n} ( t ) ^{2}}] .

The integral is local on intervals avoiding zeros of

P_{n}

. A pole of the integral at an interior polynomial zero need not be a singularity of

y

Q_{n}

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