ODE
\[ \left (a^2+x^2\right )^2 y''(x)+2 x \left (a^2+2 x^2\right ) y'(x)-y(x) \left (\text {a0}+\text {a2} x^2+\text {a4} x^4\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 4.97949 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \left (\unicode {f817}^2+a^2\right )^2+\left (-\text {a4} \unicode {f817}^4-\text {a2} \unicode {f817}^2-\text {a0}\right ) \unicode {f818}(\unicode {f817})+\left (4 \unicode {f817}^3+2 a^2 \unicode {f817}\right ) \unicode {f818}'(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✗
cpu = 0.772 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +2\,{\frac {x \left ( {a}^{2}+2\,{x}^{2} \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{ \left ( {a}^{2}+{x}^{2} \right ) ^{2}}}-{\frac { \left ( {\it a4}\,{x}^{4}+{\it a2}\,{x}^{2}+{\it a0} \right ) {\it \_Y} \left ( x \right ) }{ \left ( {a}^{2}+{x}^{2} \right ) ^{2}}} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[-((a0 + a2*x^2 + a4*x^4)*y[x]) + 2*x*(a^2 + 2*x^2)*y'[x] + (a^2 + x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(-a0 - \[FormalX]^2*a2 - \[FormalX]^4*a4)*\[FormalY][\[FormalX]] + (4*\[FormalX]^3 + 2*\[FormalX]*a^2)*Derivative[1][\[FormalY]][\[FormalX]] + (\[FormalX]^2 + a^2)^2*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
Maple raw input
dsolve((a^2+x^2)^2*diff(diff(y(x),x),x)+2*x*(a^2+2*x^2)*diff(y(x),x)-(a4*x^4+a2*x^2+a0)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = DESol({diff(diff(_Y(x),x),x)+2*x*(a^2+2*x^2)/(a^2+x^2)^2*diff(_Y(x),x)-(a4*x^4+a2*x^2+a0)/(a^2+x^2)^2*_Y(x)},{_Y(x)})