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 = 0.0237286 (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 = 0.156 (sec), leaf count = 36
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( {x}^{2}+1 \right ) ^{1-{\frac {a}{2}}}+{\it \_C2}\,{\mbox {$_2$F$_1$}(1,{\frac {a}{2}}-{\frac {1}{2}};\,{\frac {3}{2}};\,-{x}^{2})}x \right \} \] 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),'implicit')
Maple raw output
y(x) = _C1*(x^2+1)^(1-1/2*a)+_C2*hypergeom([1, 1/2*a-1/2],[3/2],-x^2)*x