\[ a x^k-(b-1) b+x^2 \left (y'(x)+y(x)^2\right )=0 \] ✓ Mathematica : cpu = 0.153984 (sec), leaf count = 821
DSolve[-((-1 + b)*b) + a*x^k + x^2*(y[x]^2 + Derivative[1][y][x]) == 0,y[x],x]
\[\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} \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \operatorname {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 (\operatorname {BesselJ}\left (\frac {2 b-1}{k}-1,\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\operatorname {BesselJ}\left (\frac {2 b-1}{k}+1,\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \operatorname {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} \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \operatorname {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 (\operatorname {BesselJ}\left (\frac {1-2 b}{k}-1,\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\operatorname {BesselJ}\left (\frac {1-2 b}{k}+1,\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \operatorname {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}} \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) c_1 \operatorname {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} \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \operatorname {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 = 0.105 (sec), leaf count = 217
dsolve(x^2*(diff(y(x),x)+y(x)^2)+a*x^k-b*(b-1) = 0,y(x))
\[y \left (x \right ) = \frac {-\operatorname {BesselJ}\left (\frac {\sqrt {\left (-1+2 b \right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \sqrt {a}\, x^{\frac {k}{2}}-\sqrt {a}\, \operatorname {BesselY}\left (\frac {\sqrt {\left (-1+2 b \right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) x^{\frac {k}{2}} c_{1}+\left (\frac {1}{2}+\left (b -\frac {1}{2}\right ) \operatorname {csgn}\left (-1+2 b \right )\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}+\operatorname {BesselJ}\left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}{x \left (\operatorname {BesselY}\left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}+\operatorname {BesselJ}\left (\frac {\sqrt {\left (-1+2 b \right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}\]