4.35.27 \(a^2 (-y(x))+\left (x^2+1\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0\)

ODE
\[ a^2 (-y(x))+\left (x^2+1\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 7.16434 (sec), leaf count = 0 , DifferentialRoot result

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

Maple ✓
cpu = 0.325 (sec), leaf count = 54

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it HeunC} \left ( 2,-{\frac {1}{2}},-{\frac {1}{2}},-1,{\frac {7}{8}}+{\frac {{a}^{2}}{4}}, \left ( {x}^{2}+1 \right ) ^{-1} \right ) +{{\it \_C2}{\it HeunC} \left ( 2,{\frac {1}{2}},-{\frac {1}{2}},-1,{\frac {7}{8}}+{\frac {{a}^{2}}{4}}, \left ( {x}^{2}+1 \right ) ^{-1} \right ) {\frac {1}{\sqrt {{x}^{2}+1}}}} \right \} \] Mathematica raw input

DSolve[-(a^2*y[x]) - 2*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]}, {-(a^2*\[FormalY][
\[FormalX]]) + (-2*\[FormalX] + 2*\[FormalX]^3)*Derivative[1][\[FormalY]][\[Form
alX]] + (1 + \[FormalX]^2)^2*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[Forma
lY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}

Maple raw input

dsolve((x^2+1)^2*diff(diff(y(x),x),x)-2*x*(-x^2+1)*diff(y(x),x)-a^2*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*HeunC(2,-1/2,-1/2,-1,7/8+1/4*a^2,1/(x^2+1))+_C2*HeunC(2,1/2,-1/2,-1,7
/8+1/4*a^2,1/(x^2+1))/(x^2+1)^(1/2)