ODE
\[ \left (a x^2+1\right ) y''(x)+a x y'(x)+b^2 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.184863 (sec), leaf count = 44
\[\left \{\left \{y(x)\to c_1 \cos \left (\frac {b \sinh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}\right )+c_2 \sin \left (\frac {b \sinh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.229 (sec), leaf count = 57
\[\left [y \left (x \right ) = \textit {\_C1} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+1}\right )^{\frac {i b}{\sqrt {a}}}+\textit {\_C2} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+1}\right )^{-\frac {i b}{\sqrt {a}}}\right ]\] Mathematica raw input
DSolve[b^2*y[x] + a*x*y'[x] + (1 + a*x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[(b*ArcSinh[Sqrt[a]*x])/Sqrt[a]] + C[2]*Sin[(b*ArcSinh[Sqrt[a]*x])/Sqrt[a]]}}
Maple raw input
dsolve((a*x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+b^2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(a^(1/2)*x+(a*x^2+1)^(1/2))^(I*b/a^(1/2))+_C2/((a^(1/2)*x+(a*x^2+1)^(1/2))^(I*b/a^(1/2)))]