ODE
\[ \text {a1} x y'(x)+\text {a2} y(x)+x^2 y''(x)=0 \] ODE Classification
[[_Emden, _Fowler]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0105266 (sec), leaf count = 57
\[\left \{\left \{y(x)\to x^{\frac {1}{2} \left (-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} \left (c_2 x^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.014 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {{\it a1}}{2}}+{\frac {1}{2}}+{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}}+{\it \_C2}\,{x}^{-{\frac {{\it a1}}{2}}+{\frac {1}{2}}-{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}} \right \} \] Mathematica raw input
DSolve[a2*y[x] + a1*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x^((1 - a1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2])/2)*(C[1] + x^Sqrt[1 - 2*a1 + a1^2 - 4*a2]*C[2])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a1*x*diff(y(x),x)+a2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*x^(-1/2*a1+1/2+1/2*(a1^2-2*a1-4*a2+1)^(1/2))+_C2*x^(-1/2*a1+1/2-1/2*(a1^2-2*a1-4*a2+1)^(1/2))