4.35.35 \(x^2 \left (a^2+x^2\right )^2 y''(x)+x \left (a^2+2 x^2\right ) y'(x)+b^2 y(x)=0\)

ODE
\[ x^2 \left (a^2+x^2\right )^2 y''(x)+x \left (a^2+2 x^2\right ) y'(x)+b^2 y(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 9.06684 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f817}^2 \unicode {f818}''(\unicode {f817}) \left (\unicode {f817}^2+a^2\right )^2+b^2 \unicode {f818}(\unicode {f817})+\left (2 \unicode {f817}^3+a^2 \unicode {f817}\right ) \unicode {f818}'(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✓
cpu = 0.494 (sec), leaf count = 253

\[ \left \{ y \left ( x \right ) ={x}^{{\frac {1}{2\,{a}^{2}} \left ( {a}^{2}+\sqrt {{a}^{4}-2\,{a}^{2}-4\,{b}^{2}+1}-1 \right ) }}{{\rm e}^{ \left ( 2\,{a}^{2}+2\,{x}^{2} \right ) ^{-1}}} \left ( {\it HeunC} \left ( {\frac {1}{2\,{a}^{2}}},{\frac {1}{2}},{\frac {1}{2\,{a}^{2}}\sqrt {{a}^{4}-2\,{a}^{2}-4\,{b}^{2}+1}},{\frac {4\,{a}^{2}+1}{8\,{a}^{4}}},{\frac {{a}^{2}-6}{8\,{a}^{2}}},{\frac {{a}^{2}}{{a}^{2}+{x}^{2}}} \right ) \left ( {a}^{2}+{x}^{2} \right ) ^{-{\frac {1}{4\,{a}^{2}} \left ( {a}^{2}+\sqrt {{a}^{4}-2\,{a}^{2}-4\,{b}^{2}+1}-1 \right ) }}{\it \_C2}+{\it HeunC} \left ( {\frac {1}{2\,{a}^{2}}},-{\frac {1}{2}},{\frac {1}{2\,{a}^{2}}\sqrt {{a}^{4}-2\,{a}^{2}-4\,{b}^{2}+1}},{\frac {4\,{a}^{2}+1}{8\,{a}^{4}}},{\frac {{a}^{2}-6}{8\,{a}^{2}}},{\frac {{a}^{2}}{{a}^{2}+{x}^{2}}} \right ) \left ( {a}^{2}+{x}^{2} \right ) ^{-{\frac {1}{4\,{a}^{2}} \left ( -{a}^{2}+\sqrt {{a}^{4}-2\,{a}^{2}-4\,{b}^{2}+1}-1 \right ) }}{\it \_C1} \right ) \right \} \] Mathematica raw input

DSolve[b^2*y[x] + x*(a^2 + 2*x^2)*y'[x] + x^2*(a^2 + x^2)^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {b^2*\[FormalY][\[
FormalX]] + (2*\[FormalX]^3 + \[FormalX]*a^2)*Derivative[1][\[FormalY]][\[Formal
X]] + \[FormalX]^2*(\[FormalX]^2 + a^2)^2*Derivative[2][\[FormalY]][\[FormalX]] 
== 0, \[FormalY][1] == C[1], Derivative[1][\[FormalY]][1] == C[2]}]][x]}}

Maple raw input

dsolve(x^2*(a^2+x^2)^2*diff(diff(y(x),x),x)+x*(a^2+2*x^2)*diff(y(x),x)+b^2*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = x^(1/2*(a^2+(a^4-2*a^2-4*b^2+1)^(1/2)-1)/a^2)*exp(1/(2*a^2+2*x^2))*(HeunC
(1/2/a^2,1/2,1/2*(a^4-2*a^2-4*b^2+1)^(1/2)/a^2,1/8/a^4*(4*a^2+1),1/8/a^2*(a^2-6)
,a^2/(a^2+x^2))*(a^2+x^2)^(-1/4*(a^2+(a^4-2*a^2-4*b^2+1)^(1/2)-1)/a^2)*_C2+HeunC
(1/2/a^2,-1/2,1/2*(a^4-2*a^2-4*b^2+1)^(1/2)/a^2,1/8/a^4*(4*a^2+1),1/8/a^2*(a^2-6
),a^2/(a^2+x^2))*(a^2+x^2)^(-1/4*(-a^2+(a^4-2*a^2-4*b^2+1)^(1/2)-1)/a^2)*_C1)