= 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, $y'=y/(x+y+c)$ becomes
$$
\frac{dx}{dy}-\frac{x}{y}=1+\frac{c}{y},\qquad y\ne0.
$$
The <integrating factor> $1/y$ gives $(x/y)'=1/y+c/y^2$, hence $x+c=y(\ln|y|+C)$. This family must retain the original denominator restriction $x+y+c\ne0$. The separate zero solution $y=0$ on intervals avoiding $x=-c$ is lost by the inversion and must be restored.
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