ODE
\[ \left (x^2+1\right ) y''(x)+x y'(x)-4 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0131415 (sec), leaf count = 25
\[\left \{\left \{y(x)\to c_1 \cosh \left (2 \sinh ^{-1}(x)\right )+i c_2 \sinh \left (2 \sinh ^{-1}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.008 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sin \left ( 2\,\arctan \left ( {\frac {x}{\sqrt {-{x}^{2}-1}}} \right ) \right ) +{\it \_C2}\,\cos \left ( 2\,\arctan \left ( {\frac {x}{\sqrt {-{x}^{2}-1}}} \right ) \right ) \right \} \] Mathematica raw input
DSolve[-4*y[x] + x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cosh[2*ArcSinh[x]] + I*C[2]*Sinh[2*ArcSinh[x]]}}
Maple raw input
dsolve((x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)-4*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*sin(2*arctan(x/(-x^2-1)^(1/2)))+_C2*cos(2*arctan(x/(-x^2-1)^(1/2)))