 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/3/c/solution

y = x^{n}

on an interval not containing zero. The coefficient of

x^{n - 2}

n (n - 1) - 3 n + 3 = n^{2} - 4 n + 3 = (n - 1) (n - 3) .

The distinct roots give

y (x) = C_{1} x + C_{2} x^{3} .

These solutions are linearly independent since their Wronskian is

2 x^{3} \neq = 0

away from zero. Both nevertheless approach zero as

x \to 0

, which will make the endpoint conditions in the next part insufficient for uniqueness.

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