ODE
\[ y'''(x)+a^3 x^3 y(x)+3 a^2 x^2 y'(x)+3 a x y''(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.165633 (sec), leaf count = 68
\[\left \{\left \{y(x)\to e^{-\frac {a x^2}{2}-\sqrt {3} \sqrt {a} x} \left (c_1 e^{\sqrt {3} \sqrt {a} x}+c_3 e^{2 \sqrt {3} \sqrt {a} x}+c_2\right )\right \}\right \}\]
Maple ✓
cpu = 0.066 (sec), leaf count = 37
\[\left [y \left (x \right ) = {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (\textit {\_C1} +\textit {\_C2} \,{\mathrm e}^{\sqrt {3}\, \sqrt {a}\, x}+\textit {\_C3} \,{\mathrm e}^{-\sqrt {3}\, \sqrt {a}\, x}\right )\right ]\] Mathematica raw input
DSolve[a^3*x^3*y[x] + 3*a^2*x^2*y'[x] + 3*a*x*y''[x] + y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^(-(Sqrt[3]*Sqrt[a]*x) - (a*x^2)/2)*(E^(Sqrt[3]*Sqrt[a]*x)*C[1] + C[2] + E^(2*Sqrt[3]*Sqrt[a]*x)*C[3])}}
Maple raw input
dsolve(diff(diff(diff(y(x),x),x),x)+3*a*x*diff(diff(y(x),x),x)+3*a^2*x^2*diff(y(x),x)+a^3*x^3*y(x) = 0, y(x))
Maple raw output
[y(x) = exp(-1/2*a*x^2)*(_C1+_C2*exp(3^(1/2)*a^(1/2)*x)+_C3*exp(-3^(1/2)*a^(1/2)*x))]