ODE
\[ y(x) (a+b x)+2 x y''(x)+y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.176547 (sec), leaf count = 115
\[\left \{\left \{y(x)\to \sqrt {x} e^{-\frac {i \sqrt {b} x}{\sqrt {2}}} \left (c_1 U\left (\frac {1}{4} \left (\frac {i \sqrt {2} a}{\sqrt {b}}+3\right ),\frac {3}{2},i \sqrt {2} \sqrt {b} x\right )+c_2 L_{\frac {1}{4} \left (-\frac {i \sqrt {2} a}{\sqrt {b}}-3\right )}^{\frac {1}{2}}\left (i \sqrt {2} \sqrt {b} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.585 (sec), leaf count = 97
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {b}\, x}{2}} \KummerM \left (\frac {i a \sqrt {2}+3 \sqrt {b}}{4 \sqrt {b}}, \frac {3}{2}, i \sqrt {2}\, \sqrt {b}\, x \right ) \sqrt {x}+\textit {\_C2} \,{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {b}\, x}{2}} \KummerU \left (\frac {i a \sqrt {2}+3 \sqrt {b}}{4 \sqrt {b}}, \frac {3}{2}, i \sqrt {2}\, \sqrt {b}\, x \right ) \sqrt {x}\right ]\] Mathematica raw input
DSolve[(a + b*x)*y[x] + y'[x] + 2*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[x]*(C[1]*HypergeometricU[(3 + (I*Sqrt[2]*a)/Sqrt[b])/4, 3/2, I*Sqrt[2]*Sqrt[b]*x] + C[2]*LaguerreL[(-3 - (I*Sqrt[2]*a)/Sqrt[b])/4, 1/2, I*Sqrt[2]*Sqrt[b]*x]))/E^((I*Sqrt[b]*x)/Sqrt[2])}}
Maple raw input
dsolve(2*x*diff(diff(y(x),x),x)+diff(y(x),x)+(b*x+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-1/2*I*2^(1/2)*b^(1/2)*x)*KummerM(1/4*(I*a*2^(1/2)+3*b^(1/2))/b^(1/2),3/2,I*2^(1/2)*b^(1/2)*x)*x^(1/2)+_C2*exp(-1/2*I*2^(1/2)*b^(1/2)*x)*KummerU(1/4*(I*a*2^(1/2)+3*b^(1/2))/b^(1/2),3/2,I*2^(1/2)*b^(1/2)*x)*x^(1/2)]