Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 13 / b / Solution

On a branch with $y\ne 0$, the given derivative is nonzero wherever the original equation is defined, so use inverse-function linearization of a first-order equation and regard $x$ as a function of $y$:The integrating factor $1/y$ givesThus the nonzero solutions are given implicitly byEach branch must also retain $x+y+2\ne 0$; at those excluded points the original right-hand side is undefined and the inverse graph cannot represent a regular $y(x)$. Differentiating the implicit equation gives $y^{\prime }=1/(\ln |y|+C+1)$, equal to the required quotient wherever these restrictions hold.

The additional solution $y=0$, on any interval avoiding $x=-2$, was lost when dividing by $y$ and must be included. It is not defined through the point $(x,y)=(-2,0)$ in the original equation.