 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/1/a/solution

μ

such that the differential expression becomes

- (p y^{'})^{'}

. Its leading coefficient requires

p = μ (1 - x^{2})

- p^{'} = 3 xμ

. Thus

(1 - x^{2}) μ^{'} - 2 xμ = - 3 xμ, \frac{μ ^{'}}{μ} = - \frac{x}{1 - x ^{2}} .

Choose the positive solution

μ = 1 - x^{2}

- [(1 - x^{2})^{3/2} y_{n}^{'}]^{'} = (n^{2} - 1) 1 - x^{2} y_{n},

with

p = (1 - x^{2})^{3/2}, q = 0, w = 1 - x^{2} .

The weight is positive on the open interval. The endpoints are singular because

p

vanishes there, but the prescribed boundedness of

y_{n}, y_{n}^{'}

 New to topics? Read the docs here!