ODE
\[ -y(x) \left (a+b x+c x^2\right )+x^2 y''(x)+x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0136535 (sec), leaf count = 104
\[\left \{\left \{y(x)\to x^{\sqrt {a}} e^{-\sqrt {c} x} \left (c_1 U\left (\frac {b}{2 \sqrt {c}}+\sqrt {a}+\frac {1}{2},2 \sqrt {a}+1,2 \sqrt {c} x\right )+c_2 L_{-\sqrt {a}-\frac {b}{2 \sqrt {c}}-\frac {1}{2}}^{2 \sqrt {a}}\left (2 \sqrt {c} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.195 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl W}_{-{\frac {b}{2}{\frac {1}{\sqrt {c}}}},\,\sqrt {a}}\left (2\,\sqrt {c}x\right )}+{\it \_C1}\,{{\sl M}_{-{\frac {b}{2}{\frac {1}{\sqrt {c}}}},\,\sqrt {a}}\left (2\,\sqrt {c}x\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] Mathematica raw input
DSolve[-((a + b*x + c*x^2)*y[x]) + x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^Sqrt[a]*(C[1]*HypergeometricU[1/2 + Sqrt[a] + b/(2*Sqrt[c]), 1 + 2*Sqrt[a], 2*Sqrt[c]*x] + C[2]*LaguerreL[-1/2 - Sqrt[a] - b/(2*Sqrt[c]), 2*Sqrt[a], 2*Sqrt[c]*x]))/E^(Sqrt[c]*x)}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-(c*x^2+b*x+a)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*WhittakerW(-1/2*b/c^(1/2),a^(1/2),2*c^(1/2)*x)+_C1*WhittakerM(-1/2*b/c^(1/2),a^(1/2),2*c^(1/2)*x))/x^(1/2)