ODE
\[ \left (a+x^4+2 x^2\right ) y(x)+4 x^3 y'(x)+4 x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.206228 (sec), leaf count = 70
\[\left \{\left \{y(x)\to \frac {e^{-\frac {x^2}{4}} x^{\frac {1}{2}-\frac {\sqrt {1-a}}{2}} \left (c_2 x^{\sqrt {1-a}}+\sqrt {1-a} c_1\right )}{\sqrt {1-a}}\right \}\right \}\]
Maple ✓
cpu = 0.111 (sec), leaf count = 47
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{\frac {1}{2}+\frac {\sqrt {1-a}}{2}} {\mathrm e}^{-\frac {x^{2}}{4}}+\textit {\_C2} \,x^{\frac {1}{2}-\frac {\sqrt {1-a}}{2}} {\mathrm e}^{-\frac {x^{2}}{4}}\right ]\] Mathematica raw input
DSolve[(a + 2*x^2 + x^4)*y[x] + 4*x^3*y'[x] + 4*x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^(1/2 - Sqrt[1 - a]/2)*(Sqrt[1 - a]*C[1] + x^Sqrt[1 - a]*C[2]))/(Sqrt[1 - a]*E^(x^2/4))}}
Maple raw input
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x^3*diff(y(x),x)+(x^4+2*x^2+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(1/2+1/2*(1-a)^(1/2))*exp(-1/4*x^2)+_C2*x^(1/2-1/2*(1-a)^(1/2))*exp(-1/4*x^2)]