ODE
\[ y(x) \left (\text {a0}+\text {a2} x^2+\text {a4} x^4\right )+\left (1-x^2\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 3.24676 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\text {a4} \unicode {f817}^4+\text {a2} \unicode {f817}^2+\text {a0}\right ) \unicode {f818}(\unicode {f817})+\left (2 \unicode {f817}^3-2 \unicode {f817}\right ) \unicode {f818}'(\unicode {f817})+\left (\unicode {f817}^4-2 \unicode {f817}^2+1\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.265 (sec), leaf count = 83
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}-1 \right ) ^{{\frac {1}{2}\sqrt {-{\it a0}-{\it a4}-{\it a2}}}} \left ( {\it HeunC} \left ( 0,{\frac {1}{2}},\sqrt {-{\it a0}-{\it a4}-{\it a2}},{\frac {{\it a4}}{4}},{\frac {1}{4}}-{\frac {{\it a0}}{4}},{x}^{2} \right ) {\it \_C2}\,x+{\it HeunC} \left ( 0,-{\frac {1}{2}},\sqrt {-{\it a0}-{\it a4}-{\it a2}},{\frac {{\it a4}}{4}},{\frac {1}{4}}-{\frac {{\it a0}}{4}},{x}^{2} \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a0 + a2*x^2 + a4*x^4)*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]}, {(a0 + \[FormalX]^2*a2 + \[FormalX]^4*a4)*\[FormalY][\[FormalX]] + (-2*\[FormalX] + 2*\[FormalX]^3)*Derivative[1][\[FormalY]][\[FormalX]] + (1 - 2*\[FormalX]^2 + \[FormalX]^4)*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)-2*x*(-x^2+1)*diff(y(x),x)+(a4*x^4+a2*x^2+a0)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x^2-1)^(1/2*(-a0-a4-a2)^(1/2))*(HeunC(0,1/2,(-a0-a4-a2)^(1/2),1/4*a4,1/4-1/4*a0,x^2)*_C2*x+HeunC(0,-1/2,(-a0-a4-a2)^(1/2),1/4*a4,1/4-1/4*a0,x^2)*_C1)