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 = 1.46383 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (-\unicode {f817} k^2-b\right ) \unicode {f818}(\unicode {f817})+(-2 \unicode {f817} a-2) \unicode {f818}'(\unicode {f817})+4 (\unicode {f817}-1) \unicode {f817} \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.198 (sec), leaf count = 66
\[ \left \{ y \left ( x \right ) = \left ( -1+x \right ) ^{{\frac {3}{2}}+{\frac {a}{2}}} \left ( \sqrt {x}{\it HeunC} \left ( 0,{\frac {1}{2}},{\frac {3}{2}}+{\frac {a}{2}},-{\frac {{k}^{2}}{4}},{\frac {a}{8}}-{\frac {b}{4}}+{\frac {5}{8}},x \right ) {\it \_C2}+{\it HeunC} \left ( 0,-{\frac {1}{2}},{\frac {3}{2}}+{\frac {a}{2}},-{\frac {{k}^{2}}{4}},{\frac {a}{8}}-{\frac {b}{4}}+{\frac {5}{8}},x \right ) {\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = (-1+x)^(3/2+1/2*a)*(x^(1/2)*HeunC(0,1/2,3/2+1/2*a,-1/4*k^2,1/8*a-1/4*b+5/8,x)*_C2+HeunC(0,-1/2,3/2+1/2*a,-1/4*k^2,1/8*a-1/4*b+5/8,x)*_C1)