\[ a x^k-(b-1) b+x^2 \left (y'(x)+y(x)^2\right )=0 \] ✓ Mathematica : cpu = 0.231431 (sec), leaf count = 821
\[\left \{\left \{y(x)\to -\frac {a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \left (\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )\right ) x^{k-1} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )-1} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) k^{1-\frac {1}{k}}+\frac {1}{2} a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )+\frac {1}{2}} x^{k-1} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )-\frac {1}{2}} \left (J_{\frac {2 b-1}{k}-1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-J_{\frac {2 b-1}{k}+1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) k^{-1/k}+c_1 \left (a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \left (\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )\right ) k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}+1} x^{k-1} J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )-1}+\frac {1}{2} a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )+\frac {1}{2}} k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} x^{k-1} \left (J_{\frac {1-2 b}{k}-1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-J_{\frac {1-2 b}{k}+1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )-\frac {1}{2}}\right )}{-a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) c_1 \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )}-a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} k^{-1/k} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )}}\right \}\right \}\] ✓ Maple : cpu = 0.129 (sec), leaf count = 219
\[y \relax (x ) = \frac {-2 \BesselJ \left (\frac {\sqrt {\left (-1+2 b \right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \sqrt {a}\, x^{\frac {k}{2}}-2 x^{\frac {k}{2}} \BesselY \left (\frac {\sqrt {\left (-1+2 b \right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \sqrt {a}\, c_{1}+2 \left (\frac {1}{2}+\left (b -\frac {1}{2}\right ) \mathrm {csgn}\left (-1+2 b \right )\right ) \left (\BesselY \left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}{2 x \left (\BesselY \left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}\]