4.34.1 \(\left (a x^2+1\right ) y''(x)+a x y'(x)+b^2 y(x)=0\)

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 = 0.0327749 (sec), leaf count = 44

\[\left \{\left \{y(x)\to c_2 \sin \left (\frac {b \sinh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}\right )+c_1 \cos \left (\frac {b \sinh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}\right )\right \}\right \}\]

Maple
cpu = 0.015 (sec), leaf count = 59

\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C1}\, \left ( \left ( \sqrt {a}x+\sqrt {a{x}^{2}+1} \right ) ^{{ib{\frac {1}{\sqrt {a}}}}} \right ) ^{2}+{\it \_C2} \right ) \left ( \left ( \sqrt {a}x+\sqrt {a{x}^{2}+1} \right ) ^{{ib{\frac {1}{\sqrt {a}}}}} \right ) ^{-1}} \right \} \] 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),'implicit')

Maple raw output

y(x) = (_C1*((a^(1/2)*x+(a*x^2+1)^(1/2))^(I*b/a^(1/2)))^2+_C2)/((a^(1/2)*x+(a*x^
2+1)^(1/2))^(I*b/a^(1/2)))