ODE
\[ 4 (1-x) x (1-a x) y''(x)+y'(x) \left (\text {a0}+\text {a1} x+\text {a2} x^2\right )+y(x) \left (\text {b0}+\text {b1} x^2\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 9.11495 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\text {b1} \unicode {f817}^2+\text {b0}\right ) \unicode {f818}(\unicode {f817})+\left (\text {a2} \unicode {f817}^2+\text {a1} \unicode {f817}+\text {a0}\right ) \unicode {f818}'(\unicode {f817})+4 (\unicode {f817}-1) \unicode {f817} (\unicode {f817} a-1) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✗
cpu = 1.031 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac { \left ( {\it a2}\,{x}^{2}+{\it a1}\,x+{\it a0} \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{4\,x \left ( 1-x \right ) \left ( -ax+1 \right ) }}+{\frac { \left ( {\it b1}\,{x}^{2}+{\it b0} \right ) {\it \_Y} \left ( x \right ) }{4\,x \left ( 1-x \right ) \left ( -ax+1 \right ) }} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[(b0 + b1*x^2)*y[x] + (a0 + a1*x + a2*x^2)*y'[x] + 4*(1 - x)*x*(1 - a*x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(b0 + \[FormalX]^2*b1)*\[FormalY][\[FormalX]] + (a0 + \[FormalX]*a1 + \[FormalX]^2*a2)*Derivative[1][\[FormalY]][\[FormalX]] + 4*(-1 + \[FormalX])*\[FormalX]*(-1 + \[FormalX]*a)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][2] == C[1], Derivative[1][\[FormalY]][2] == C[2]}]][x]}}
Maple raw input
dsolve(4*x*(1-x)*(-a*x+1)*diff(diff(y(x),x),x)+(a2*x^2+a1*x+a0)*diff(y(x),x)+(b1*x^2+b0)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = DESol({diff(diff(_Y(x),x),x)+1/4*(a2*x^2+a1*x+a0)/x/(1-x)/(-a*x+1)*diff(_Y(x),x)+1/4*(b1*x^2+b0)/x/(1-x)/(-a*x+1)*_Y(x)},{_Y(x)})