ODE
\[ \left (a+b x^2\right ) y'(x)+(a-1) x (a+b) y(x)+x \left (1-x^2\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.421832 (sec), leaf count = 53
\[\left \{\left \{y(x)\to \frac {c_2 \, _2F_1\left (\frac {a-1}{2},\frac {1}{2} (-a-b);\frac {a+1}{2};x^2\right )}{a-1}+c_1 x^{1-a}\right \}\right \}\]
Maple ✓
cpu = 0.625 (sec), leaf count = 39
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{1-a}+\textit {\_C2} \hypergeom \left (\left [\frac {a}{2}-\frac {1}{2}, -\frac {a}{2}-\frac {b}{2}\right ], \left [\frac {a}{2}+\frac {1}{2}\right ], x^{2}\right )\right ]\] Mathematica raw input
DSolve[(-1 + a)*(a + b)*x*y[x] + (a + b*x^2)*y'[x] + x*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x^(1 - a)*C[1] + (C[2]*Hypergeometric2F1[(-1 + a)/2, (-a - b)/2, (1 + a)/2, x^2])/(-1 + a)}}
Maple raw input
dsolve(x*(-x^2+1)*diff(diff(y(x),x),x)+(b*x^2+a)*diff(y(x),x)+(a+b)*(a-1)*x*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(1-a)+_C2*hypergeom([1/2*a-1/2, -1/2*a-1/2*b],[1/2*a+1/2],x^2)]