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 = 0.0102824 (sec), leaf count = 49
\[\left \{\left \{y(x)\to c_1 \cosh \left (\sqrt {2} \sqrt {a} \sqrt {x}\right )+i c_2 \sinh \left (\sqrt {2} \sqrt {a} \sqrt {x}\right )\right \}\right \}\]
Maple ✓
cpu = 0.013 (sec), leaf count = 31
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sinh \left ( \sqrt {x}\sqrt {2}\sqrt {a} \right ) +{\it \_C2}\,\cosh \left ( \sqrt {x}\sqrt {2}\sqrt {a} \right ) \right \} \] Mathematica raw input
DSolve[-(a*y[x]) + y'[x] + 2*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cosh[Sqrt[2]*Sqrt[a]*Sqrt[x]] + I*C[2]*Sinh[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*sinh(x^(1/2)*2^(1/2)*a^(1/2))+_C2*cosh(x^(1/2)*2^(1/2)*a^(1/2))