Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?

Question

The special form of Riccati equation $$ \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} $$ has been proved that it is solvable if and only if $m=0$, $m=-2$, $m=4k/(2k\pm 1)$.
The sufficiency is obviously. But how to prove its necessity?

Answer 1

For $m\neq -2$ the solution contains the Bessel function $J_{\nu}\left(\frac{2 \sqrt{a b}\, x^{1+m/2}}{m+2}\right)$ with index $\nu=\pm(m+2)^{-1}$, $\nu=-1\pm(m+2)^{-1}$, and $\nu=1\pm(m+2)^{-1}$. The Bessel function becomes an elementary function (sine or cosine) if the index is half-integer, which requires $m=-4k/(2k\pm 1)$ at integer $k$.