 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/4/c/solution

Let

A (x) = \sum_{n = 1}^{\infty} a_{n} x^{n}

denote the analytic part of the prescribed ansatz. Its logarithmic part has derivatives

\frac{d}{d x} [(b + c x) ln x] = c ln x + \frac{b}{x} + c, \frac{d ^{2}}{d x ^{2}} [(b + c x) ln x] = \frac{c}{x} - \frac{b}{x ^{2}} .

After substitution, the full differential equation reduces to

x (1 - x) A^{''} + (1 - x) A^{'} + A + 2 c (1 - x) + (b + c) ln x = 0.

The first terms are real analytic at zero and cannot cancel a nonzero logarithm. Therefore

c = - b .

The cancellation of the apparent

1/ x

terms is already built into this calculation.

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