ODE
\[ a x^2 y'(x)+x^2 y''(x)-2 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0207021 (sec), leaf count = 80
\[\left \{\left \{y(x)\to -\frac {a x^{3/2} e^{-\frac {a x}{2}} \left (2 \left (2 c_1+i a c_2 x\right ) \sinh \left (\frac {a x}{2}\right )-2 \left (a c_1 x+2 i c_2\right ) \cosh \left (\frac {a x}{2}\right )\right )}{\sqrt {\pi } (-i a x)^{5/2}}\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 28
\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C2}\, \left ( ax+2 \right ) {{\rm e}^{-ax}}+{\it \_C1}\, \left ( ax-2 \right ) }{x}} \right \} \] Mathematica raw input
DSolve[-2*y[x] + a*x^2*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((a*x^(3/2)*(-2*(a*x*C[1] + (2*I)*C[2])*Cosh[(a*x)/2] + 2*(2*C[1] + I*a*x*C[2])*Sinh[(a*x)/2]))/(E^((a*x)/2)*Sqrt[Pi]*((-I)*a*x)^(5/2)))}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*x^2*diff(y(x),x)-2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*(a*x+2)*exp(-a*x)+_C1*(a*x-2))/x