 Solution

For the Bernoulli differential equation, work on a branch where $y\ne 0$ and the required powers are defined. Multiplication by $(1-m)y^{-m}$ makes the chain rule useful: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.