##### 4.14.21 $$\left (x^2+1\right ) \left (y(x)^2+1\right ) y'(x)+2 x y(x) \left (1-y(x)^2\right )=0$$

ODE
$\left (x^2+1\right ) \left (y(x)^2+1\right ) y'(x)+2 x y(x) \left (1-y(x)^2\right )=0$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.479069 (sec), leaf count = 83

$\left \{\left \{y(x)\to \frac {1}{2} \left (-e^{c_1} \left (x^2+1\right )-\sqrt {4+e^{2 c_1} \left (x^2+1\right )^2}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {4+e^{2 c_1} \left (x^2+1\right )^2}-e^{c_1} \left (x^2+1\right )\right )\right \}\right \}$

Maple
cpu = 0.219 (sec), leaf count = 75

$\left [y \left (x \right ) = \frac {x^{2} \textit {\_C1}}{2}+\frac {\textit {\_C1}}{2}-\frac {\sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1}^{2} x^{2}+\textit {\_C1}^{2}+4}}{2}, y \left (x \right ) = \frac {x^{2} \textit {\_C1}}{2}+\frac {\textit {\_C1}}{2}+\frac {\sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1}^{2} x^{2}+\textit {\_C1}^{2}+4}}{2}\right ]$ Mathematica raw input

DSolve[2*x*y[x]*(1 - y[x]^2) + (1 + x^2)*(1 + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-(E^C[1]*(1 + x^2)) - Sqrt[4 + E^(2*C[1])*(1 + x^2)^2])/2}, {y[x] ->
(-(E^C[1]*(1 + x^2)) + Sqrt[4 + E^(2*C[1])*(1 + x^2)^2])/2}}

Maple raw input

dsolve((x^2+1)*(1+y(x)^2)*diff(y(x),x)+2*x*y(x)*(1-y(x)^2) = 0, y(x))

Maple raw output

[y(x) = 1/2*x^2*_C1+1/2*_C1-1/2*(_C1^2*x^4+2*_C1^2*x^2+_C1^2+4)^(1/2), y(x) = 1/
2*x^2*_C1+1/2*_C1+1/2*(_C1^2*x^4+2*_C1^2*x^2+_C1^2+4)^(1/2)]