 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/8/c/solution

For

ℓ = 0

the terminating even solution is constant; for

ℓ = 1

the terminating odd solution is proportional to

x

. For

ℓ = 2

, the recurrence gives

a_{2} = - 3 a_{0}

; for

ℓ = 3

it gives

a_{3} = - 5 a_{1} /3

. Imposing the Legendre polynomial normalization at

1

yields

P_{0} (x) = 1, P_{1} (x) = x, P_{2} (x) = \frac{3 x ^{2} - 1}{2}, P_{3} (x) = \frac{5 x ^{3} - 3 x}{2} .

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