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 = 1.35619 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to (x)\right \}\right \}\]

Maple ✓
cpu = 2.284 (sec), leaf count = 54

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

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)]