ODE
\[ a x \left (1-x^2\right ) y'(x)+b y(x)+\left (x^2+1\right )^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 1.92653 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \left (\unicode {f817}^2+1\right )^2+b \unicode {f818}(\unicode {f817})+\left (\unicode {f817} a-\unicode {f817}^3 a\right ) \unicode {f818}'(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.344 (sec), leaf count = 101
\[ \left \{ y \left ( x \right ) ={{\rm e}^{{\frac {a}{{x}^{2}+1}}}} \left ( {\it HeunC} \left ( a,-{\frac {1}{2}}-{\frac {a}{2}},-{\frac {1}{2}},a+{\frac {{a}^{2}}{4}},-{\frac {7\,a}{8}}-{\frac {{a}^{2}}{4}}+{\frac {1}{8}}-{\frac {b}{4}}, \left ( {x}^{2}+1 \right ) ^{-1} \right ) \left ( {x}^{2}+1 \right ) ^{{\frac {1}{2}}+{\frac {a}{2}}}{\it \_C2}+{\it HeunC} \left ( a,{\frac {1}{2}}+{\frac {a}{2}},-{\frac {1}{2}},a+{\frac {{a}^{2}}{4}},-{\frac {7\,a}{8}}-{\frac {{a}^{2}}{4}}+{\frac {1}{8}}-{\frac {b}{4}}, \left ( {x}^{2}+1 \right ) ^{-1} \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[b*y[x] + a*x*(1 - x^2)*y'[x] + (1 + x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {b*\[FormalY][\[FormalX]] + (\[FormalX]*a - \[FormalX]^3*a)*Derivative[1][\[FormalY]][\[FormalX]] + (1 + \[FormalX]^2)^2*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
Maple raw input
dsolve((x^2+1)^2*diff(diff(y(x),x),x)+a*x*(-x^2+1)*diff(y(x),x)+b*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(a/(x^2+1))*(HeunC(a,-1/2-1/2*a,-1/2,a+1/4*a^2,-7/8*a-1/4*a^2+1/8-1/4*b,1/(x^2+1))*(x^2+1)^(1/2+1/2*a)*_C2+HeunC(a,1/2+1/2*a,-1/2,a+1/4*a^2,-7/8*a-1/4*a^2+1/8-1/4*b,1/(x^2+1))*_C1)