4.36.3 \(-y(x) \left (a (a+1) (1-x)+b^2 x\right )+4 \left (1-x^2\right ) x^2 y''(x)+2 (1-3 x) (1-x) x y'(x)=0\)

ODE
\[ -y(x) \left (a (a+1) (1-x)+b^2 x\right )+4 \left (1-x^2\right ) x^2 y''(x)+2 (1-3 x) (1-x) x y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 1.88493 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (-\unicode {f817} a^2+a^2-\unicode {f817} a+a+\unicode {f817} b^2\right ) \unicode {f818}(\unicode {f817})-2 (\unicode {f817}-1) \unicode {f817} (3 \unicode {f817}-1) \unicode {f818}'(\unicode {f817})+\left (4 \unicode {f817}^4-4 \unicode {f817}^2\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(2)=c_1,\unicode {f818}'(2)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✓
cpu = 0.347 (sec), leaf count = 93

\[ \left \{ y \left ( x \right ) = \left ( 1+x \right ) ^{3} \left ( {\it \_C2}\,{\it HeunG} \left ( -1,{\frac {{a}^{2}}{4}}-{\frac {{b}^{2}}{4}}+{\frac {9\,a}{4}}+{\frac {7}{2}},{\frac {a}{2}}+1,{\frac {a}{2}}+{\frac {7}{2}},{\frac {3}{2}}+a,0,x \right ) {x}^{{\frac {1}{2}}+{\frac {a}{2}}}+{\it \_C1}\,{\it HeunG} \left ( -1,{\frac {{a}^{2}}{4}}-{\frac {{b}^{2}}{4}}-{\frac {7\,a}{4}}+{\frac {3}{2}},-{\frac {a}{2}}+3,-{\frac {a}{2}}+{\frac {1}{2}},{\frac {1}{2}}-a,0,x \right ) {x}^{-{\frac {a}{2}}} \right ) \right \} \] Mathematica raw input

DSolve[-((a*(1 + a)*(1 - x) + b^2*x)*y[x]) + 2*(1 - 3*x)*(1 - x)*x*y'[x] + 4*x^2*(1 - x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(a - \[FormalX]*a
 + a^2 - \[FormalX]*a^2 + \[FormalX]*b^2)*\[FormalY][\[FormalX]] - 2*(-1 + \[For
malX])*\[FormalX]*(-1 + 3*\[FormalX])*Derivative[1][\[FormalY]][\[FormalX]] + (-
4*\[FormalX]^2 + 4*\[FormalX]^4)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[F
ormalY][2] == C[1], Derivative[1][\[FormalY]][2] == C[2]}]][x]}}

Maple raw input

dsolve(4*x^2*(-x^2+1)*diff(diff(y(x),x),x)+2*x*(1-x)*(1-3*x)*diff(y(x),x)-(a*(1+a)*(1-x)+b^2*x)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = (1+x)^3*(_C2*HeunG(-1,1/4*a^2-1/4*b^2+9/4*a+7/2,1/2*a+1,1/2*a+7/2,3/2+a,0
,x)*x^(1/2+1/2*a)+_C1*HeunG(-1,1/4*a^2-1/4*b^2-7/4*a+3/2,-1/2*a+3,-1/2*a+1/2,1/2
-a,0,x)*x^(-1/2*a))