ODE
\[ \left (a^2-x^2\right )^2 y''(x)-2 x \left (a^2-x^2\right ) y'(x)+y(x) \left (\text {a0}+\text {a2} x^2+\text {a4} x^4\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 5.33723 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(a-\unicode {f817})^2 \unicode {f818}''(\unicode {f817}) (\unicode {f817}+a)^2-2 \unicode {f817} (a-\unicode {f817}) \unicode {f818}'(\unicode {f817}) (\unicode {f817}+a)+\left (\text {a4} \unicode {f817}^4+\text {a2} \unicode {f817}^2+\text {a0}\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.306 (sec), leaf count = 144
\[ \left \{ y \left ( x \right ) = \left ( \left ( a-x \right ) \left ( a+x \right ) \right ) ^{{\frac {1}{2\,a}\sqrt {-{\it a4}\,{a}^{4}-{\it a2}\,{a}^{2}-{\it a0}}}} \left ( {\it HeunC} \left ( 0,{\frac {1}{2}},{\frac {1}{a}\sqrt {-{\it a4}\,{a}^{4}-{\it a2}\,{a}^{2}-{\it a0}}},{\frac {{a}^{2}{\it a4}}{4}},{\frac {{a}^{2}-{\it a0}}{4\,{a}^{2}}},{\frac {{x}^{2}}{{a}^{2}}} \right ) {\it \_C2}\,x+{\it HeunC} \left ( 0,-{\frac {1}{2}},{\frac {1}{a}\sqrt {-{\it a4}\,{a}^{4}-{\it a2}\,{a}^{2}-{\it a0}}},{\frac {{a}^{2}{\it a4}}{4}},{\frac {{a}^{2}-{\it a0}}{4\,{a}^{2}}},{\frac {{x}^{2}}{{a}^{2}}} \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a0 + a2*x^2 + a4*x^4)*y[x] - 2*x*(a^2 - x^2)*y'[x] + (a^2 - 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]*(-\[FormalX] + a)*(\[FormalX] + a)*Derivative[1][\[FormalY]][\[FormalX]] + (-\[FormalX] + a)^2*(\[FormalX] + a)^2*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
Maple raw input
dsolve((a^2-x^2)^2*diff(diff(y(x),x),x)-2*x*(a^2-x^2)*diff(y(x),x)+(a4*x^4+a2*x^2+a0)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = ((a-x)*(a+x))^(1/2*(-a^4*a4-a^2*a2-a0)^(1/2)/a)*(HeunC(0,1/2,(-a^4*a4-a^2*a2-a0)^(1/2)/a,1/4*a^2*a4,1/4/a^2*(a^2-a0),x^2/a^2)*_C2*x+HeunC(0,-1/2,(-a^4*a4-a^2*a2-a0)^(1/2)/a,1/4*a^2*a4,1/4/a^2*(a^2-a0),x^2/a^2)*_C1)