4.35.5 \(y(x) (a+b x)+4 (1-x) x y''(x)+2 (1-3 x) (1-x) y'(x)=0\)

ODE
\[ y(x) (a+b x)+4 (1-x) x y''(x)+2 (1-3 x) (1-x) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

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

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(-a-\unicode {f817} b) \unicode {f818}(\unicode {f817})+\left (-6 \unicode {f817}^2+8 \unicode {f817}-2\right ) \unicode {f818}'(\unicode {f817})+4 (\unicode {f817}-1) \unicode {f817} \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \},\langle \langle \rangle \rangle \right )(x)\right \}\right \}\]

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

\[ \left \{ y \left ( x \right ) = \left ( -1+x \right ) \left ( {\it \_C2}\,{\it HeunC} \left ( -{\frac {3}{2}},{\frac {1}{2}},1,{\frac {3}{8}}-{\frac {b}{4}},{\frac {1}{8}}-{\frac {a}{4}},x \right ) \sqrt {x}+{\it \_C1}\,{\it HeunC} \left ( -{\frac {3}{2}},-{\frac {1}{2}},1,{\frac {3}{8}}-{\frac {b}{4}},{\frac {1}{8}}-{\frac {a}{4}},x \right ) \right ) \right \} \] 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))