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 = 1.9852 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to (x)\right \}\right \}\]

Maple ✓
cpu = 1.6 (sec), leaf count = 97

\[\left [y \left (x \right ) = \textit {\_C1} \left (x +1\right )^{3} \mathit {HG}\left (-1, \frac {1}{4} a^{2}-\frac {1}{4} b^{2}-\frac {7}{4} a +\frac {3}{2}, 3-\frac {a}{2}, -\frac {a}{2}+\frac {1}{2}, \frac {1}{2}-a , 0, x\right ) x^{-\frac {a}{2}}+\textit {\_C2} \left (x +1\right )^{3} \mathit {HG}\left (-1, \frac {1}{4} a^{2}-\frac {1}{4} b^{2}+\frac {9}{4} a +\frac {7}{2}, \frac {a}{2}+1, \frac {7}{2}+\frac {a}{2}, \frac {3}{2}+a , 0, x\right ) x^{\frac {a}{2}+\frac {1}{2}}\right ]\] 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))

Maple raw output

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