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 = 1.70313 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to (x)\right \}\right \}\]
Maple ✓
cpu = 1.675 (sec), leaf count = 64
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{\frac {\sqrt {c}\, x}{2}} \HeunC \left (\sqrt {c}, -\frac {1}{2}, -\frac {1}{2}, -\frac {b}{4}-\frac {c}{4}, -\frac {a}{4}+\frac {3}{8}, x\right )+\textit {\_C2} \,{\mathrm e}^{\frac {\sqrt {c}\, x}{2}} \HeunC \left (\sqrt {c}, \frac {1}{2}, -\frac {1}{2}, -\frac {b}{4}-\frac {c}{4}, -\frac {a}{4}+\frac {3}{8}, x\right ) \sqrt {x}\right ]\] 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))
Maple raw output
[y(x) = _C1*exp(1/2*c^(1/2)*x)*HeunC(c^(1/2),-1/2,-1/2,-1/4*b-1/4*c,-1/4*a+3/8,x)+_C2*exp(1/2*c^(1/2)*x)*HeunC(c^(1/2),1/2,-1/2,-1/4*b-1/4*c,-1/4*a+3/8,x)*x^(1/2)]