\[ y(x) \left (a x^k+(1-b) b\right )+x^2 y''(x)=0 \] ✓ Mathematica : cpu = 0.0361444 (sec), leaf count = 225
DSolve[((1 - b)*b + a*x^k)*y[x] + x^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_2 k^{-1/k} a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right \}\right \}\] ✓ Maple : cpu = 0.044 (sec), leaf count = 67
dsolve(x^2*diff(diff(y(x),x),x)+(a*x^k-b*(b-1))*y(x)=0,y(x))
\[y \left (x \right ) = \sqrt {x}\, \left (\BesselY \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{2}+\BesselJ \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}\right )\]