4.33.41 \(y(x) (a+b x)+4 (1-x) x y''(x)+2 (1-2 x) y'(x)=0\)

ODE
\[ y(x) (a+b x)+4 (1-x) x y''(x)+2 (1-2 x) y'(x)=0 \] ODE Classification

[_Jacobi]

Book solution method
TO DO

Mathematica ✗
cpu = 1.0983 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(-a-\unicode {f817} b) \unicode {f818}(\unicode {f817})+(4 \unicode {f817}-2) \unicode {f818}'(\unicode {f817})+4 (\unicode {f817}-1) \unicode {f817} \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✓
cpu = 0.142 (sec), leaf count = 35

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it MathieuC} \left ( {\frac {b}{2}}+a,-{\frac {b}{4}},\arccos \left ( \sqrt {x} \right ) \right ) +{\it \_C2}\,{\it MathieuS} \left ( {\frac {b}{2}}+a,-{\frac {b}{4}},\arccos \left ( \sqrt {x} \right ) \right ) \right \} \] Mathematica raw input

DSolve[(a + b*x)*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)*\[FormalY][\[FormalX]] + (-2 + 4*\[FormalX])*Derivative[1][\[FormalY]][\[Form
alX]] + 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)+(b*x+a)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*MathieuC(1/2*b+a,-1/4*b,arccos(x^(1/2)))+_C2*MathieuS(1/2*b+a,-1/4*b,
arccos(x^(1/2)))