ODE
\[ y(x) \left (a+b x^2\right )+x^2 y''(x)+x^2 y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.185231 (sec), leaf count = 85
\[\left \{\left \{y(x)\to e^{-x/2} \sqrt {x} \left (c_1 J_{\frac {1}{2} \sqrt {1-4 a}}\left (-\frac {1}{2} i \sqrt {1-4 b} x\right )+c_2 Y_{\frac {1}{2} \sqrt {1-4 a}}\left (-\frac {1}{2} i \sqrt {1-4 b} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.107 (sec), leaf count = 63
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {x}{2}} \sqrt {x}\, \BesselJ \left (\frac {\sqrt {1-4 a}}{2}, \frac {\sqrt {4 b -1}\, x}{2}\right )+\textit {\_C2} \,{\mathrm e}^{-\frac {x}{2}} \sqrt {x}\, \BesselY \left (\frac {\sqrt {1-4 a}}{2}, \frac {\sqrt {4 b -1}\, x}{2}\right )\right ]\] Mathematica raw input
DSolve[(a + b*x^2)*y[x] + x^2*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[x]*(BesselJ[Sqrt[1 - 4*a]/2, (-1/2*I)*Sqrt[1 - 4*b]*x]*C[1] + BesselY[Sqrt[1 - 4*a]/2, (-1/2*I)*Sqrt[1 - 4*b]*x]*C[2]))/E^(x/2)}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+(b*x^2+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-1/2*x)*x^(1/2)*BesselJ(1/2*(1-4*a)^(1/2),1/2*(4*b-1)^(1/2)*x)+_C2*exp(-1/2*x)*x^(1/2)*BesselY(1/2*(1-4*a)^(1/2),1/2*(4*b-1)^(1/2)*x)]