4.32.5 $ax{y}^{\prime }(x)-(2-a)y(x)+(x^2+1)y^{″}(x)=0$

ODE
$$ax{y}^{\prime }(x)-(2-a)y(x)+(x^2+1)y^{″}(x)=0$$ ODE Classification

[[_2nd_order, _exact, _linear, _homogeneous]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0237286 (sec), leaf count = 68

$$\left\{\{y(x)\to {(x^2+1)}^{\frac{1}{2}-\frac{a}{4}}(c_1{P}_{\frac{a-4}{2}}^{\frac{a-2}{2}}(ix)+c_2{Q}_{\frac{a-4}{2}}^{\frac{a-2}{2}}(ix))\}\right\}$$

Maple ✓
cpu = 0.156 (sec), leaf count = 36

$$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{(x^2+1)}^{1-\frac{a}{2}}+\mathit{\_}\mathit{C}\mathit{2}{}_2\text{F}{}_1(1,\frac{a}{2}-\frac{1}{2};\frac{3}{2};-x^2)x\}$$ 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