ODE
\[ \left (x^2+1\right )^2 y''(x)+2 x \left (x^2+1\right ) y'(x)+4 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.159897 (sec), leaf count = 22
\[\left \{\left \{y(x)\to c_1 \cos \left (2 \tan ^{-1}(x)\right )+c_2 \sin \left (2 \tan ^{-1}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.043 (sec), leaf count = 19
\[[y \left (x \right ) = \textit {\_C1} \sin \left (2 \arctan \left (x \right )\right )+\textit {\_C2} \cos \left (2 \arctan \left (x \right )\right )]\] Mathematica raw input
DSolve[4*y[x] + 2*x*(1 + x^2)*y'[x] + (1 + x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[2*ArcTan[x]] + C[2]*Sin[2*ArcTan[x]]}}
Maple raw input
dsolve((x^2+1)^2*diff(diff(y(x),x),x)+2*x*(x^2+1)*diff(y(x),x)+4*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sin(2*arctan(x))+_C2*cos(2*arctan(x))]