Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 8 / a

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- Solution a

Solution

Divide by

2 x^{2}

to put the ordinary differential equation into normalized form

y^{''} + P (x) y^{'} + Q (x) y = 0

:

P (x) = - \frac{1}{2 x}, Q (x) = \frac{1 + x}{2 x ^{2}} .

The coefficients are singular at zero, but

x P (x) = - \frac{1}{2}, x^{2} Q (x) = \frac{1 + x}{2}

are both analytic there. This is precisely the regular singular point criterion for a second-order equation, so $x = 0$ is a regular singular point.

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