ODE
\[ a x y(x)+x y''(x)+2 y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0286669 (sec), leaf count = 51
\[\left \{\left \{y(x)\to \frac {e^{-\sqrt {-a} x} \left (\frac {c_2 e^{2 \sqrt {-a} x}}{\sqrt {-a}}+2 c_1\right )}{2 x}\right \}\right \}\]
Maple ✓
cpu = 0.048 (sec), leaf count = 29
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C1}\,\sinh \left ( \sqrt {-a}x \right ) +{\it \_C2}\,\cosh \left ( \sqrt {-a}x \right ) \right ) } \right \} \] Mathematica raw input
DSolve[a*x*y[x] + 2*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2*C[1] + (E^(2*Sqrt[-a]*x)*C[2])/Sqrt[-a])/(2*E^(Sqrt[-a]*x)*x)}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+2*diff(y(x),x)+a*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C1*sinh((-a)^(1/2)*x)+_C2*cosh((-a)^(1/2)*x))/x