4.45.45 $4ax{y}^{‴}(x)+y^{⁗}(x)+a^4{x}^{4}y(x)+4a^3{x}^3{y}^{\prime }(x)+6a^2{x}^2{y}^{″}(x)=0$

ODE
$$4ax{y}^{‴}(x)+y^{⁗}(x)+a^4{x}^{4}y(x)+4a^3{x}^3{y}^{\prime }(x)+6a^2{x}^2{y}^{″}(x)=0$$ ODE Classification

[[_high_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.721502 (sec), leaf count = 148

$$\left\{\{y(x)\to \frac{e^{-\frac{1}{2}x(ax+2\sqrt{-(\sqrt{6}-3)a})}(6\sqrt{a}(c_2{e}^{2\sqrt{-(\sqrt{6}-3)a}x}+c_1)+\sqrt{18-6\sqrt{6}}{e}^{-\frac{(-3+\sqrt{3}+\sqrt{6})ax}{\sqrt{-(\sqrt{6}-3)a}}}(c_4{e}^{\frac{2ax}{\sqrt{a-\sqrt{\frac{2}{3}}a}}}+c_3))}{6\sqrt{a}}\}\right\}$$

Maple ✓
cpu = 0.046 (sec), leaf count = 84

$$\{y\left(x\right)-{\mathrm{e}}^{-\frac{a{x}^2}{2}}(\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{-\sqrt{-a\sqrt{6}+3a}x}+\mathit{\_}\mathit{C}\mathit{2}{\mathrm{e}}^{\sqrt{-a\sqrt{6}+3a}x}+\mathit{\_}\mathit{C}\mathit{3}{\mathrm{e}}^{-\sqrt{a\sqrt{6}+3a}x}+\mathit{\_}\mathit{C}\mathit{4}{\mathrm{e}}^{\sqrt{a\sqrt{6}+3a}x})=0\}$$ 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] -> (6*Sqrt[a]*(C[1] + E^(2*Sqrt[-((-3 + Sqrt[6])*a)]*x)*C[2]) + (Sqrt[18 
- 6*Sqrt[6]]*(C[3] + E^((2*a*x)/Sqrt[a - Sqrt[2/3]*a])*C[4]))/E^(((-3 + Sqrt[3] 
+ Sqrt[6])*a*x)/Sqrt[-((-3 + Sqrt[6])*a)]))/(6*Sqrt[a]*E^((x*(2*Sqrt[-((-3 + Sqr
t[6])*a)] + a*x))/2))}}

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),'implicit')

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)) =
 0