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 = 0.166798 (sec), leaf count = 46
\[\left \{\left \{y(x)\to c_1 \cos \left (\sqrt {2} \sqrt {a} \sqrt {x}\right )+c_2 \sin \left (\sqrt {2} \sqrt {a} \sqrt {x}\right )\right \}\right \}\]
Maple ✓
cpu = 0.089 (sec), leaf count = 31
\[\left [y \left (x \right ) = \textit {\_C1} \sin \left (\sqrt {x}\, \sqrt {2}\, \sqrt {a}\right )+\textit {\_C2} \cos \left (\sqrt {x}\, \sqrt {2}\, \sqrt {a}\right )\right ]\] 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))
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))]