ODE
\[ \left (a^2-x^2\right )^2 \left (b^2-x^2\right ) y''(x)+x \left (\text {a0}+\text {b0} x^2\right ) y'(x)+y(x) \left (\text {a1}+\text {b1} x^2+\text {c1} x^4\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 122.993 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{-(a-\unicode {f817})^2 (b-\unicode {f817}) (\unicode {f817}+b) \unicode {f818}''(\unicode {f817}) (\unicode {f817}+a)^2+\left (-\text {c1} \unicode {f817}^4-\text {b1} \unicode {f817}^2-\text {a1}\right ) \unicode {f818}(\unicode {f817})-\unicode {f817} \left (\text {b0} \unicode {f817}^2+\text {a0}\right ) \unicode {f818}'(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✗
cpu = 2.188 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac {x \left ( {x}^{2}{\it b0}+{\it a0} \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{ \left ( {a}^{2}-{x}^{2} \right ) ^{2} \left ( {b}^{2}-{x}^{2} \right ) }}+{\frac { \left ( {\it c1}\,{x}^{4}+{\it b1}\,{x}^{2}+{\it a1} \right ) {\it \_Y} \left ( x \right ) }{ \left ( {a}^{2}-{x}^{2} \right ) ^{2} \left ( {b}^{2}-{x}^{2} \right ) }} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[(a1 + b1*x^2 + c1*x^4)*y[x] + x*(a0 + b0*x^2)*y'[x] + (a^2 - x^2)^2*(b^2 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(-a1 - \[FormalX]^2*b1 - \[FormalX]^4*c1)*\[FormalY][\[FormalX]] - \[FormalX]*(a0 + \[FormalX]^2*b0)*Derivative[1][\[FormalY]][\[FormalX]] - (-\[FormalX] + a)^2*(\[FormalX] + a)^2*(-\[FormalX] + b)*(\[FormalX] + b)*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*(b^2-x^2)*diff(diff(y(x),x),x)+x*(b0*x^2+a0)*diff(y(x),x)+(c1*x^4+b1*x^2+a1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = DESol({diff(diff(_Y(x),x),x)+x*(b0*x^2+a0)/(a^2-x^2)^2/(b^2-x^2)*diff(_Y(x),x)+(c1*x^4+b1*x^2+a1)/(a^2-x^2)^2/(b^2-x^2)*_Y(x)},{_Y(x)})