 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/d/solution

Let

W = P_{ℓ} Q_{ℓ}^{'} - P_{ℓ}^{'} Q_{ℓ}

W^{'} = P_{ℓ} Q_{ℓ}^{''} - P_{ℓ}^{''} Q_{ℓ} = \frac{2 x}{1 - x ^{2}} (P_{ℓ} Q_{ℓ}^{'} - P_{ℓ}^{'} Q_{ℓ}) = \frac{2 x}{1 - x ^{2}} W .

Thus

[(1 - x^{2}) W]^{'} = 0

, so

W (x) = \frac{A _{ℓ}}{1 - x ^{2}},

with

A_{ℓ}

independent of

x

on each interval avoiding the singular endpoints. This is the Abel identity. Linear independence forces

A_{ℓ} \neq = 0

, and its value depends on the chosen normalization of

Q_{ℓ}

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