ODE
\[ -y(x) \left (a^2+b^2 x^2+c^2 x^4\right )+x^2 y''(x)+x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.12044 (sec), leaf count = 96
\[\left \{\left \{y(x)\to \frac {2^{\frac {a+1}{2}} \left (x^2\right )^{\frac {a+1}{2}} e^{-\frac {c x^2}{2}} \left (c_1 U\left (\frac {b^2+2 (a+1) c}{4 c},a+1,c x^2\right )+c_2 L_{-\frac {2 (a+1) c+b^2}{4 c}}^a\left (c x^2\right )\right )}{x}\right \}\right \}\]
Maple ✓
cpu = 0.114 (sec), leaf count = 47
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C2}\,{{\sl W}_{-{\frac {{b}^{2}}{4\,c}},\,{\frac {a}{2}}}\left (c{x}^{2}\right )}+{\it \_C1}\,{{\sl M}_{-{\frac {{b}^{2}}{4\,c}},\,{\frac {a}{2}}}\left (c{x}^{2}\right )} \right ) } \right \} \] Mathematica raw input
DSolve[-((a^2 + b^2*x^2 + c^2*x^4)*y[x]) + x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^((1 + a)/2)*(x^2)^((1 + a)/2)*(C[1]*HypergeometricU[(b^2 + 2*(1 + a)*c)/(4*c), 1 + a, c*x^2] + C[2]*LaguerreL[-(b^2 + 2*(1 + a)*c)/(4*c), a, c*x^2]))/(E^((c*x^2)/2)*x)}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-(c^2*x^4+b^2*x^2+a^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*WhittakerW(-1/4/c*b^2,1/2*a,c*x^2)+_C1*WhittakerM(-1/4/c*b^2,1/2*a,c*x^2))/x