##### 4.45.45 $$4 a x y'''(x)+y''''(x)+a^4 x^4 y(x)+4 a^3 x^3 y'(x)+6 a^2 x^2 y''(x)=0$$

ODE
$4 a x y'''(x)+y''''(x)+a^4 x^4 y(x)+4 a^3 x^3 y'(x)+6 a^2 x^2 y''(x)=0$ ODE Classiﬁcation

[[_high_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.712797 (sec), leaf count = 165

$\left \{\left \{y(x)\to \frac {e^{-\frac {a x^2}{2}-\sqrt {3+\sqrt {6}} \sqrt {a} x} \left (6 a \left (c_1 e^{\frac {\left (-3+\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}+c_2 e^{\frac {\left (3+\sqrt {3}-\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}\right )+\sqrt {6} \sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )} \left (c_4 e^{\frac {2 a x}{\sqrt {a-\sqrt {\frac {2}{3}} a}}}+c_3\right )\right )}{6 a}\right \}\right \}$

Maple
cpu = 0.204 (sec), leaf count = 81

$\left [y \left (x \right ) = {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (\textit {\_C1} \,{\mathrm e}^{-\sqrt {-a \sqrt {6}+3 a}\, x}+\textit {\_C2} \,{\mathrm e}^{\sqrt {-a \sqrt {6}+3 a}\, x}+\textit {\_C3} \,{\mathrm e}^{-\sqrt {a \sqrt {6}+3 a}\, x}+\textit {\_C4} \,{\mathrm e}^{\sqrt {a \sqrt {6}+3 a}\, x}\right )\right ]$ Mathematica raw input

DSolve[a^4*x^4*y[x] + 4*a^3*x^3*y'[x] + 6*a^2*x^2*y''[x] + 4*a*x*y'''[x] + y''''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (E^(-(Sqrt[3 + Sqrt[6]]*Sqrt[a]*x) - (a*x^2)/2)*(6*a*(E^(((-3 + Sqrt[3
] + Sqrt[6])*a*x)/Sqrt[-((-3 + Sqrt[6])*a)])*C[1] + E^(((3 + Sqrt[3] - Sqrt[6])*
a*x)/Sqrt[-((-3 + Sqrt[6])*a)])*C[2]) + Sqrt[6]*Sqrt[-((-3 + Sqrt[6])*a)]*(C[3]
+ E^((2*a*x)/Sqrt[a - Sqrt[2/3]*a])*C[4])))/(6*a)}}

Maple raw input

dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+4*a*x*diff(diff(diff(y(x),x),x),x)+6*a^2*x^2*diff(diff(y(x),x),x)+4*a^3*x^3*diff(y(x),x)+a^4*x^4*y(x) = 0, y(x))

Maple raw output

[y(x) = exp(-1/2*a*x^2)*(_C1*exp(-(-a*6^(1/2)+3*a)^(1/2)*x)+_C2*exp((-a*6^(1/2)+
3*a)^(1/2)*x)+_C3*exp(-(a*6^(1/2)+3*a)^(1/2)*x)+_C4*exp((a*6^(1/2)+3*a)^(1/2)*x)
)]