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 = 0.306053 (sec), leaf count = 61

\[\left \{\left \{y(x)\to c_2 \sqrt {x} \text {HeunC}\left [\frac {a-3}{4},\frac {1}{4} (-b-3),\frac {3}{2},0,-\frac {3}{2},x\right ]+c_1 \text {HeunC}\left [\frac {a}{4},-\frac {b}{4},\frac {1}{2},0,-\frac {3}{2},x\right ]\right \}\right \}\]

Maple ✓
cpu = 1.141 (sec), leaf count = 48

\[\left [y \left (x \right ) = \textit {\_C1} \HeunC \left (-\frac {3}{2}, -\frac {1}{2}, 1, \frac {3}{8}-\frac {b}{4}, \frac {1}{8}-\frac {a}{4}, x\right ) \left (x -1\right )+\textit {\_C2} \sqrt {x}\, \HeunC \left (-\frac {3}{2}, \frac {1}{2}, 1, \frac {3}{8}-\frac {b}{4}, \frac {1}{8}-\frac {a}{4}, x\right ) \left (x -1\right )\right ]\] 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] -> Sqrt[x]*C[2]*HeunC[(-3 + a)/4, (-3 - b)/4, 3/2, 0, -3/2, x] + C[1]*Heu
nC[a/4, -1/4*b, 1/2, 0, -3/2, 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))

Maple raw output

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