ODE
\[ -\left (1-2 a x^3\right ) y'(x)+a x^2 \left (a x^3+1\right ) y(x)+x y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.175122 (sec), leaf count = 30
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-\frac {a x^3}{3}} \left (c_2 x^2+2 c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.073 (sec), leaf count = 26
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {a \,x^{3}}{3}}+\textit {\_C2} \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} x^{2}\right ]\] Mathematica raw input
DSolve[a*x^2*(1 + a*x^3)*y[x] - (1 - 2*a*x^3)*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2*C[1] + x^2*C[2])/(2*E^((a*x^3)/3))}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)-(-2*a*x^3+1)*diff(y(x),x)+a*(a*x^3+1)*x^2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-1/3*a*x^3)+_C2*exp(-1/3*a*x^3)*x^2]