 Inverse-function linearization of a first-order equation

On a branch where a differentiable function $y(x)$ has a nonzero derivative, the inverse function theorem allows the ordinary differential equation to be rewritten for $x(y)$. A nonlinear equation in one direction can become a linear ordinary differential equation in the other. For example, $y^{\prime }=y/(x+y+c)$ becomesThe integrating factor $1/y$ gives $(x/y{)}^{\prime }=1/y+c/y^2$, hence $x+c=y(\ln |y|+C)$. This family must retain the original denominator restriction $x+y+c\ne 0$. The separate zero solution $y=0$ on intervals avoiding $x=-c$ is lost by the inversion and must be restored.