ODE
\[ a^2 y(x)+\left (x^2+1\right ) y''(x)+x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.16339 (sec), leaf count = 22
\[\left \{\left \{y(x)\to c_1 \cos \left (a \sinh ^{-1}(x)\right )+c_2 \sin \left (a \sinh ^{-1}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.067 (sec), leaf count = 19
\[[y \left (x \right ) = \textit {\_C1} \sin \left (a \arcsinh \left (x \right )\right )+\textit {\_C2} \cos \left (a \arcsinh \left (x \right )\right )]\] Mathematica raw input
DSolve[a^2*y[x] + x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[a*ArcSinh[x]] + C[2]*Sin[a*ArcSinh[x]]}}
Maple raw input
dsolve((x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+a^2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sin(a*arcsinh(x))+_C2*cos(a*arcsinh(x))]