ODE
\[ a y(x)+2 x y''(x)+y'(x)=0 \] ODE Classification
[[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0097279 (sec), leaf count = 46
\[\left \{\left \{y(x)\to c_2 \sin \left (\sqrt {2} \sqrt {a} \sqrt {x}\right )+c_1 \cos \left (\sqrt {2} \sqrt {a} \sqrt {x}\right )\right \}\right \}\]
Maple ✓
cpu = 0.014 (sec), leaf count = 31
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sin \left ( \sqrt {x}\sqrt {2}\sqrt {a} \right ) +{\it \_C2}\,\cos \left ( \sqrt {x}\sqrt {2}\sqrt {a} \right ) \right \} \] Mathematica raw input
DSolve[a*y[x] + y'[x] + 2*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[Sqrt[2]*Sqrt[a]*Sqrt[x]] + C[2]*Sin[Sqrt[2]*Sqrt[a]*Sqrt[x]]}}
Maple raw input
dsolve(2*x*diff(diff(y(x),x),x)+diff(y(x),x)+a*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*sin(x^(1/2)*2^(1/2)*a^(1/2))+_C2*cos(x^(1/2)*2^(1/2)*a^(1/2))