ODE
\[ y(x) \left (a+b x+c x^2\right )+4 (1-x) x y''(x)+2 (1-2 x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 1.68152 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (-c \unicode {f817}^2-b \unicode {f817}-a\right ) \unicode {f818}(\unicode {f817})+(4 \unicode {f817}-2) \unicode {f818}'(\unicode {f817})+4 (\unicode {f817}-1) \unicode {f817} \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.307 (sec), leaf count = 58
\[ \left \{ y \left ( x \right ) ={{\rm e}^{{\frac {x}{2}\sqrt {c}}}} \left ( \sqrt {x}{\it HeunC} \left ( \sqrt {c},{\frac {1}{2}},-{\frac {1}{2}},-{\frac {b}{4}}-{\frac {c}{4}},-{\frac {a}{4}}+{\frac {3}{8}},x \right ) {\it \_C2}+{\it HeunC} \left ( \sqrt {c},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {b}{4}}-{\frac {c}{4}},-{\frac {a}{4}}+{\frac {3}{8}},x \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a + b*x + c*x^2)*y[x] + 2*(1 - 2*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 - \[FormalX]^2*c)*\[FormalY][\[FormalX]] + (-2 + 4*\[FormalX])*Derivative[1][\[FormalY]][\[FormalX]] + 4*(-1 + \[FormalX])*\[FormalX]*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][2] == C[1], Derivative[1][\[FormalY]][2] == C[2]}]][x]}}
Maple raw input
dsolve(4*x*(1-x)*diff(diff(y(x),x),x)+2*(1-2*x)*diff(y(x),x)+(c*x^2+b*x+a)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(1/2*c^(1/2)*x)*(x^(1/2)*HeunC(c^(1/2),1/2,-1/2,-1/4*b-1/4*c,-1/4*a+3/8,x)*_C2+HeunC(c^(1/2),-1/2,-1/2,-1/4*b-1/4*c,-1/4*a+3/8,x)*_C1)