ODE
\[ \left (a-(a+1) x^2\right ) y'(x)+c x y(x)+x \left (1-x^2\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.222322 (sec), leaf count = 130
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (a-\sqrt {a^2+4 c}\right ),\frac {1}{4} \left (a+\sqrt {a^2+4 c}\right );\frac {a+1}{2};x^2\right )+i^{1-a} c_2 x^{1-a} \, _2F_1\left (\frac {1}{4} \left (-a-\sqrt {a^2+4 c}+2\right ),\frac {1}{4} \left (-a+\sqrt {a^2+4 c}+2\right );\frac {3-a}{2};x^2\right )\right \}\right \}\]
Maple ✓
cpu = 0.126 (sec), leaf count = 73
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {a}{2}}+{\frac {1}{2}}} \left ( {\it LegendreQ} \left ( -{\frac {1}{2}}+{\frac {1}{2}\sqrt {{a}^{2}+4\,c}},{\frac {a}{2}}-{\frac {1}{2}},\sqrt {-{x}^{2}+1} \right ) {\it \_C2}+{\it LegendreP} \left ( -{\frac {1}{2}}+{\frac {1}{2}\sqrt {{a}^{2}+4\,c}},{\frac {a}{2}}-{\frac {1}{2}},\sqrt {-{x}^{2}+1} \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[c*x*y[x] + (a - (1 + a)*x^2)*y'[x] + x*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> I^(1 - a)*x^(1 - a)*C[2]*Hypergeometric2F1[(2 - a - Sqrt[a^2 + 4*c])/4, (2 - a + Sqrt[a^2 + 4*c])/4, (3 - a)/2, x^2] + C[1]*Hypergeometric2F1[(a - Sqrt[a^2 + 4*c])/4, (a + Sqrt[a^2 + 4*c])/4, (1 + a)/2, x^2]}}
Maple raw input
dsolve(x*(-x^2+1)*diff(diff(y(x),x),x)+(a-(1+a)*x^2)*diff(y(x),x)+c*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(-1/2*a+1/2)*(LegendreQ(-1/2+1/2*(a^2+4*c)^(1/2),1/2*a-1/2,(-x^2+1)^(1/2))*_C2+LegendreP(-1/2+1/2*(a^2+4*c)^(1/2),1/2*a-1/2,(-x^2+1)^(1/2))*_C1)