4.35.6 $4(1-x)x(1-ax)y^{″}(x)+y^{\prime }(x)(\text{a0}+\text{a1}x+\text{a2}{x}^2)+y(x)(\text{b0}+\text{b1}{x}^2)=0$

ODE
$$4(1-x)x(1-ax)y^{″}(x)+y^{\prime }(x)(\text{a0}+\text{a1}x+\text{a2}{x}^2)+y(x)(\text{b0}+\text{b1}{x}^2)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 9.11495 (sec), leaf count = 0 , DifferentialRoot result

$$\text{left {left {y(x)to text {DifferentialRoot}left ({unicode {f818},unicode {f817}}unicode {f4a1}left {left (text {b1} unicode {f817}^2+text {b0}right ) unicode {f818}(unicode {f817})+left (text {a2} unicode {f817}^2+text {a1} unicode {f817}+text {a0}right ) unicode {f818}'(unicode {f817})+4 (unicode {f817}-1) unicode {f817} (unicode {f817} a-1) unicode {f818}''(unicode {f817})=0,unicode {f818}(2)=c\_1,unicode {f818}'(2)=c\_2right }right )(x)right }right }}$$

Maple ✗
cpu = 1.031 (sec), leaf count = 0 , result contains DESol

$$\{y\left(x\right)=\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}(\{\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\mathit{\_}\mathit{Y}\left(x\right)+\frac{(\mathit{a}\mathit{2}{x}^2+\mathit{a}\mathit{1}x+\mathit{a}\mathit{0})\frac{\mathrm{d}}{\mathrm{d}x}\mathit{\_}\mathit{Y}\left(x\right)}{4x(1-x)(-ax+1)}+\frac{(\mathit{b}\mathit{1}{x}^2+\mathit{b}\mathit{0})\mathit{\_}\mathit{Y}\left(x\right)}{4x(1-x)(-ax+1)}\},\left\{\mathit{\_}\mathit{Y}\left(x\right)\right\})\}$$

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)})