\[ a y'(x)+b x^{\text {a1}} y(x)+x y''(x)=0 \] ✓ Mathematica : cpu = 0.0359605 (sec), leaf count = 441
DSolve[b*x^a1*y[x] + a*Derivative[1][y][x] + x*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (\frac {1}{\text {a1}}+1\right )^{\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} \text {a1}^{\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} b^{\frac {1}{2} \left (\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \left (x^{\text {a1}}\right )^{\frac {1}{2} \left (\frac {1}{\text {a1}}+1\right ) \left (\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \Gamma \left (\frac {a}{\text {a1}+1}+\frac {\text {a1}}{\text {a1}+1}\right ) J_{\frac {a-1}{\text {a1}+1}}\left (\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {1}{2} \left (1+\frac {1}{\text {a1}}\right )}}{\left (1+\frac {1}{\text {a1}}\right ) \text {a1}}\right )+c_2 \left (\frac {1}{\text {a1}}+1\right )^{-\frac {2 (1-a)}{\text {a1}+1}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}+\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} \text {a1}^{-\frac {2 (1-a)}{\text {a1}+1}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}+\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} b^{\frac {1-a}{\text {a1}+1}+\frac {1}{2} \left (\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \left (x^{\text {a1}}\right )^{\frac {1}{2} \left (\frac {1}{\text {a1}}+1\right ) \left (\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )+\frac {(1-a) \left (\frac {1}{\text {a1}}+1\right )}{\text {a1}+1}} \Gamma \left (-\frac {a}{\text {a1}+1}+\frac {\text {a1}}{\text {a1}+1}+\frac {2}{\text {a1}+1}\right ) J_{\frac {1-a}{\text {a1}+1}}\left (\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {1}{2} \left (1+\frac {1}{\text {a1}}\right )}}{\left (1+\frac {1}{\text {a1}}\right ) \text {a1}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.093 (sec), leaf count = 71
dsolve(x*diff(diff(y(x),x),x)+a*diff(y(x),x)+b*x^a1*y(x)=0,y(x))
\[y \left (x \right ) = x^{-\frac {a}{2}+\frac {1}{2}} \left (\BesselY \left (\frac {a -1}{\mathit {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\mathit {a1}}{2}+\frac {1}{2}}}{\mathit {a1} +1}\right ) c_{2}+\BesselJ \left (\frac {a -1}{\mathit {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\mathit {a1}}{2}+\frac {1}{2}}}{\mathit {a1} +1}\right ) c_{1}\right )\]