4.35.5 $y(x)(a+bx)+4(1-x)x{y}^{″}(x)+2(1-3x)(1-x)y^{\prime }(x)=0$

ODE
$$y(x)(a+bx)+4(1-x)x{y}^{″}(x)+2(1-3x)(1-x)y^{\prime }(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 0.954187 (sec), leaf count = 0 , DifferentialRoot result

$$\text{left {left {y(x)to text {DifferentialRoot}left ({unicode {f818},unicode {f817}}unicode {f4a1}left {(-a-unicode {f817} b) unicode {f818}(unicode {f817})+left (-6 unicode {f817}^2+8 unicode {f817}-2right ) unicode {f818}'(unicode {f817})+4 (unicode {f817}-1) unicode {f817} unicode {f818}''(unicode {f817})=0,unicode {f818}(2)=c\_1,unicode {f818}'(2)=c\_2right },langle langle rangle rangle right )(x)right }right }}$$

Maple ✓
cpu = 0.184 (sec), leaf count = 46

$$\{y\left(x\right)=(-1+x)(\mathit{\_}\mathit{C}\mathit{2}\mathit{H}\mathit{e}\mathit{u}\mathit{n}\mathit{C}(-\frac{3}{2},\frac{1}{2},1,\frac{3}{8}-\frac{b}{4},\frac{1}{8}-\frac{a}{4},x)\sqrt{x}+\mathit{\_}\mathit{C}\mathit{1}\mathit{H}\mathit{e}\mathit{u}\mathit{n}\mathit{C}(-\frac{3}{2},-\frac{1}{2},1,\frac{3}{8}-\frac{b}{4},\frac{1}{8}-\frac{a}{4},x))\}$$ Mathematica raw input

DSolve[(a + b*x)*y[x] + 2*(1 - 3*x)*(1 - x)*y'[x] + 4*(1 - x)*x*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(-a - \[FormalX]*
b)*\[FormalY][\[FormalX]] + (-2 + 8*\[FormalX] - 6*\[FormalX]^2)*Derivative[1][\
[FormalY]][\[FormalX]] + 4*(-1 + \[FormalX])*\[FormalX]*Derivative[2][\[FormalY]
][\[FormalX]] == 0, \[FormalY][2] == C[1], Derivative[1][\[FormalY]][2] == C[2]}
], Function[\[FormalX], {{Re[\[FormalX]] <= 0, Im[\[FormalX]] == 0}}]][x]}}

Maple raw input

dsolve(4*x*(1-x)*diff(diff(y(x),x),x)+2*(1-x)*(1-3*x)*diff(y(x),x)+(b*x+a)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = (-1+x)*(_C2*HeunC(-3/2,1/2,1,3/8-1/4*b,1/8-1/4*a,x)*x^(1/2)+_C1*HeunC(-3/
2,-1/2,1,3/8-1/4*b,1/8-1/4*a,x))