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 = 4.04825 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to (x)\right \}\right \}\]
Maple ✓
cpu = 1.609 (sec), leaf count = 176
\[\left [y \left (x \right ) = \textit {\_C1} \left (\left (a -x \right ) \left (a +x \right )\right )^{\frac {\sqrt {-a^{4} \mathit {a4} -a^{2} \mathit {a2} -\mathit {a0}}}{2 a}} \HeunC \left (0, -\frac {1}{2}, \frac {\sqrt {-a^{4} \mathit {a4} -a^{2} \mathit {a2} -\mathit {a0}}}{a}, \frac {a^{2} \mathit {a4}}{4}, \frac {a^{2}-\mathit {a0}}{4 a^{2}}, \frac {x^{2}}{a^{2}}\right )+\textit {\_C2} \left (\left (a -x \right ) \left (a +x \right )\right )^{\frac {\sqrt {-a^{4} \mathit {a4} -a^{2} \mathit {a2} -\mathit {a0}}}{2 a}} x \HeunC \left (0, \frac {1}{2}, \frac {\sqrt {-a^{4} \mathit {a4} -a^{2} \mathit {a2} -\mathit {a0}}}{a}, \frac {a^{2} \mathit {a4}}{4}, \frac {a^{2}-\mathit {a0}}{4 a^{2}}, \frac {x^{2}}{a^{2}}\right )\right ]\] 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))
Maple raw output
[y(x) = _C1*((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),1/a^2*x^2)+_C2*((a-x)*(a+x))^(1/2*(-a^4*a4-a^2*a2-a0)^(1/2)/a)*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),1/a^2*x^2)]