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 = 0.721502 (sec), leaf count = 148
\[\left \{\left \{y(x)\to \frac {e^{-\frac {1}{2} x \left (a x+2 \sqrt {-\left (\sqrt {6}-3\right ) a}\right )} \left (6 \sqrt {a} \left (c_2 e^{2 \sqrt {-\left (\sqrt {6}-3\right ) a} x}+c_1\right )+\sqrt {18-6 \sqrt {6}} e^{-\frac {\left (-3+\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\sqrt {6}-3\right ) a}}} \left (c_4 e^{\frac {2 a x}{\sqrt {a-\sqrt {\frac {2}{3}} a}}}+c_3\right )\right )}{6 \sqrt {a}}\right \}\right \}\]
Maple ✓
cpu = 0.046 (sec), leaf count = 84
\[ \left \{ y \left ( x \right ) -{{\rm e}^{-{\frac {a{x}^{2}}{2}}}} \left ( {\it \_C1}\,{{\rm e}^{-\sqrt {-a\sqrt {6}+3\,a}x}}+{\it \_C2}\,{{\rm e}^{\sqrt {-a\sqrt {6}+3\,a}x}}+{\it \_C3}\,{{\rm e}^{-\sqrt {a\sqrt {6}+3\,a}x}}+{\it \_C4}\,{{\rm e}^{\sqrt {a\sqrt {6}+3\,a}x}} \right ) =0 \right \} \] 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] -> (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 + Sqrt[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