ODE
\[ a x^2 y(x)+x y''(x)+2 y'(x)=0 \] ODE Classification
[[_Emden, _Fowler]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0111032 (sec), leaf count = 36
\[\left \{\left \{y(x)\to \frac {c_1 \text {Ai}\left (\sqrt [3]{-a} x\right )+c_2 \text {Bi}\left (\sqrt [3]{-a} x\right )}{x}\right \}\right \}\]
Maple ✓
cpu = 0.072 (sec), leaf count = 33
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl Y}_{{\frac {1}{3}}}\left ({\frac {2}{3}\sqrt {a}{x}^{{\frac {3}{2}}}}\right )}+{\it \_C1}\,{{\sl J}_{{\frac {1}{3}}}\left ({\frac {2}{3}\sqrt {a}{x}^{{\frac {3}{2}}}}\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] Mathematica raw input
DSolve[a*x^2*y[x] + 2*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (AiryAi[(-a)^(1/3)*x]*C[1] + AiryBi[(-a)^(1/3)*x]*C[2])/x}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+2*diff(y(x),x)+a*x^2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*BesselY(1/3,2/3*a^(1/2)*x^(3/2))+_C1*BesselJ(1/3,2/3*a^(1/2)*x^(3/2)))/x^(1/2)