Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 11 / c

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 $u^{\prime }=-y^{\prime }/y^2$ andAn integrating factor is $1/x$ on either side of zero. ThusThe full solution set isUse connected intervals with $x\ne 0$ and, in the nonzero family, $Cx-x^2\ne 0$. The separate zero solution would be lost by dividing by $y^2$.