 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit/11/c/solution

The zero function is a solution on any interval avoiding

x = 0

. For a nonzero solution, the Bernoulli differential equation substitution

u = 1/ y

gives

u^{'} = - y^{'} / y^{2}

and

u^{'} - \frac{u}{x} = - x .

1/ x

on either side of zero. Thus

(\frac{u}{x})^{'} = - 1, \frac{u}{x} = C - x, u = C x - x^{2} .

y (x) = \frac{1}{C x - x ^{2}}, or y (x) = 0.

Use connected intervals with

x \neq = 0

and, in the nonzero family,

C x - x^{2} \neq = 0

. The separate zero solution would be lost by dividing by

y^{2}

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