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 = 0.308194 (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 = 0.599 (sec), leaf count = 79
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {a}{2}+\frac {1}{2}} \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {a^{2}+4 c}}{2}, \frac {a}{2}-\frac {1}{2}, \sqrt {-x^{2}+1}\right )+\textit {\_C2} \,x^{-\frac {a}{2}+\frac {1}{2}} \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {a^{2}+4 c}}{2}, \frac {a}{2}-\frac {1}{2}, \sqrt {-x^{2}+1}\right )\right ]\] 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))
Maple raw output
[y(x) = _C1*x^(-1/2*a+1/2)*LegendreP(-1/2+1/2*(a^2+4*c)^(1/2),1/2*a-1/2,(-x^2+1)^(1/2))+_C2*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))]