ODE
\[ 2 (a x+1) y'(x)+y(x) \left (b+k^2 x\right )+4 (1-x) x y''(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✗
cpu = 1.47387 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to (x)\right \}\right \}\]
Maple ✓
cpu = 1.252 (sec), leaf count = 74
\[\left [y \left (x \right ) = \textit {\_C1} \left (x -1\right )^{\frac {3}{2}+\frac {a}{2}} \HeunC \left (0, -\frac {1}{2}, \frac {3}{2}+\frac {a}{2}, -\frac {k^{2}}{4}, \frac {a}{8}-\frac {b}{4}+\frac {5}{8}, x\right )+\textit {\_C2} \sqrt {x}\, \left (x -1\right )^{\frac {3}{2}+\frac {a}{2}} \HeunC \left (0, \frac {1}{2}, \frac {3}{2}+\frac {a}{2}, -\frac {k^{2}}{4}, \frac {a}{8}-\frac {b}{4}+\frac {5}{8}, x\right )\right ]\] Mathematica raw input
DSolve[(b + k^2*x)*y[x] + 2*(1 + a*x)*y'[x] + 4*(1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(-b - \[FormalX]*k^2)*\[FormalY][\[FormalX]] + (-2 - 2*\[FormalX]*a)*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*(a*x+1)*diff(y(x),x)+(k^2*x+b)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x-1)^(3/2+1/2*a)*HeunC(0,-1/2,3/2+1/2*a,-1/4*k^2,1/8*a-1/4*b+5/8,x)+_C2*x^(1/2)*(x-1)^(3/2+1/2*a)*HeunC(0,1/2,3/2+1/2*a,-1/4*k^2,1/8*a-1/4*b+5/8,x)]