ODE
\[ \left (a+b x^2\right ) y'(x)+2 (1-b) x y(x)+x \left (1-x^2\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.258774 (sec), leaf count = 106
\[\left \{\left \{y(x)\to \frac {x^{-a} \left (a+(b-1) x^2+1\right ) \left ((a-1) (a+1)^2 c_1 x^a-c_2 x F_1\left (\frac {1-a}{2};\frac {1}{2} (-a-b),2;\frac {3-a}{2};x^2,-\frac {(b-1) x^2}{a+1}\right )\right )}{(a-1) (a+1)^2 (a+4 b-3)}\right \}\right \}\]
Maple ✓
cpu = 0.329 (sec), leaf count = 50
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( {x}^{2} \left ( -1+b \right ) +1+a \right ) +{\it \_C2}\,{x}^{1-a}{\mbox {$_2$F$_1$}(-{\frac {1}{2}}-{\frac {a}{2}},1-{\frac {b}{2}}-{\frac {a}{2}};\,-{\frac {a}{2}}+{\frac {3}{2}};\,{x}^{2})} \right \} \] Mathematica raw input
DSolve[2*(1 - 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] -> ((1 + a + (-1 + b)*x^2)*((-1 + a)*(1 + a)^2*x^a*C[1] - x*AppellF1[(1 - a)/2, (-a - b)/2, 2, (3 - a)/2, x^2, -(((-1 + b)*x^2)/(1 + a))]*C[2]))/((-1 + a)*(1 + a)^2*(-3 + a + 4*b)*x^a)}}
Maple raw input
dsolve(x*(-x^2+1)*diff(diff(y(x),x),x)+(b*x^2+a)*diff(y(x),x)+2*(1-b)*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*(x^2*(-1+b)+1+a)+_C2*x^(1-a)*hypergeom([-1/2-1/2*a, 1-1/2*b-1/2*a],[-1/2*a+3/2],x^2)