4.36.8 \(x \left (a^2-x^2\right ) \left (b^2-x^2\right ) y''(x)+\left (\text {a0}+\text {b0} x^4\right ) y'(x)+x^3 y(x) \left (\text {a1}+\text {b1} x^2+\text {c1} x^4\right )=0\)

ODE
\[ x \left (a^2-x^2\right ) \left (b^2-x^2\right ) y''(x)+\left (\text {a0}+\text {b0} x^4\right ) y'(x)+x^3 y(x) \left (\text {a1}+\text {b1} x^2+\text {c1} x^4\right )=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 96.9451 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\text {c1} \unicode {f817}^7+\text {b1} \unicode {f817}^5+\text {a1} \unicode {f817}^3\right ) \unicode {f818}(\unicode {f817})+\left (\text {b0} \unicode {f817}^4+\text {a0}\right ) \unicode {f818}'(\unicode {f817})+\unicode {f817} (a-\unicode {f817}) (\unicode {f817}+a) (b-\unicode {f817}) (\unicode {f817}+b) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✗
cpu = 1.983 (sec), leaf count = 0 , result contains DESol

\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac { \left ( {\it b0}\,{x}^{4}+{\it a0} \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{x \left ( {a}^{2}-{x}^{2} \right ) \left ( {b}^{2}-{x}^{2} \right ) }}+{\frac {{x}^{2} \left ( {\it c1}\,{x}^{4}+{\it b1}\,{x}^{2}+{\it a1} \right ) {\it \_Y} \left ( x \right ) }{ \left ( {a}^{2}-{x}^{2} \right ) \left ( {b}^{2}-{x}^{2} \right ) }} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]

Mathematica raw input

DSolve[x^3*(a1 + b1*x^2 + c1*x^4)*y[x] + (a0 + b0*x^4)*y'[x] + x*(a^2 - x^2)*(b^2 - x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(\[FormalX]^3*a1 
+ \[FormalX]^5*b1 + \[FormalX]^7*c1)*\[FormalY][\[FormalX]] + (a0 + \[FormalX]^4
*b0)*Derivative[1][\[FormalY]][\[FormalX]] + \[FormalX]*(-\[FormalX] + a)*(\[For
malX] + a)*(-\[FormalX] + b)*(\[FormalX] + b)*Derivative[2][\[FormalY]][\[Formal
X]] == 0, \[FormalY][1] == C[1], Derivative[1][\[FormalY]][1] == C[2]}]][x]}}

Maple raw input

dsolve(x*(a^2-x^2)*(b^2-x^2)*diff(diff(y(x),x),x)+(b0*x^4+a0)*diff(y(x),x)+x^3*(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)+(b0*x^4+a0)/x/(a^2-x^2)/(b^2-x^2)*diff(_Y(x)
,x)+x^2*(c1*x^4+b1*x^2+a1)/(a^2-x^2)/(b^2-x^2)*_Y(x)},{_Y(x)})