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

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- Solution 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

: the product extends by reduction of order across a zero of the known solution. After a standard normalization the independent second solution is the Legendre function of the second kind

Q_{n}

.

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