\[ y''(x)=-\frac {y(x) \left (\left (x^2-1\right ) \left (a x^2+b x+c\right )-k^2\right )}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✗ Mathematica : cpu = 4.41111 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (a \unicode {f817}^4+b \unicode {f817}^3-a \unicode {f817}^2+c \unicode {f817}^2-b \unicode {f817}-k^2-c\right ) \unicode {f818}(\unicode {f817})+\left (2 \unicode {f817}^3-2 \unicode {f817}\right ) \unicode {f818}'(\unicode {f817})+\left (\unicode {f817}^4-2 \unicode {f817}^2+1\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
✓ Maple : cpu = 0.572 (sec), leaf count = 110
\[ \left \{ y \left ( x \right ) ={{\rm e}^{\sqrt {-a}x}} \left ( \sqrt {2\,x-2} \left ( 1+x \right ) ^{-{\frac {k}{2}}} \left ( x-1 \right ) ^{{\frac {k}{2}}-{\frac {1}{2}}}{\it HeunC} \left ( 4\,\sqrt {-a},-k,k,2\,b,{\frac {{k}^{2}}{2}}+a-b+c,{\frac {1}{2}}+{\frac {x}{2}} \right ) {\it \_C2}+ \left ( {x}^{2}-1 \right ) ^{{\frac {k}{2}}}{\it HeunC} \left ( 4\,\sqrt {-a},k,k,2\,b,{\frac {{k}^{2}}{2}}+a-b+c,{\frac {1}{2}}+{\frac {x}{2}} \right ) {\it \_C1} \right ) \right \} \]