ODE
\[ a x y'(x)-((2-a) y(x))+\left (x^2+1\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _exact, _linear, _homogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.174338 (sec), leaf count = 68
\[\left \{\left \{y(x)\to \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (c_1 P_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)+c_2 Q_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.519 (sec), leaf count = 36
\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}+1\right )^{-\frac {a}{2}+1}+\textit {\_C2} \hypergeom \left (\left [1, \frac {a}{2}-\frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) x\right ]\] Mathematica raw input
DSolve[-((2 - a)*y[x]) + a*x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (1 + x^2)^(1/2 - a/4)*(C[1]*LegendreP[(-4 + a)/2, (-2 + a)/2, I*x] + C[2]*LegendreQ[(-4 + a)/2, (-2 + a)/2, I*x])}}
Maple raw input
dsolve((x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)-(2-a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x^2+1)^(-1/2*a+1)+_C2*hypergeom([1, 1/2*a-1/2],[3/2],-x^2)*x]