= Solution
For the <Bernoulli differential equation>, work on a branch where $y\ne0$ and the required powers are defined. Multiplication by $(1-m)y^{-m}$ makes the <chain rule> useful:
$$
z=y^{1-m},\qquad
\boxed{z'+(1-m)f(x)z=(1-m)g(x).}
$$
Thus $\alpha=1-m$, $h=(1-m)f$ and $k=(1-m)g$. The case $m=1$ is already linear and must be treated separately, because the proposed substitution would be constant. Solutions lost by dividing by a power of $y$, particularly $y=0$ when it solves the original equation, must also be checked separately.
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