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 = 0.0531684 (sec), leaf count = 83
\[\left \{\left \{y(x)\to \frac {1}{2} \left (-e^{c_1} \left (x^2+1\right )-\sqrt {e^{2 c_1} \left (x^2+1\right )^2+4}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {e^{2 c_1} \left (x^2+1\right )^2+4}-e^{c_1} \left (x^2+1\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.025 (sec), leaf count = 28
\[ \left \{ {\it \_C1}+\ln \left ( {x}^{2}+1 \right ) -\ln \left ( y \left ( x \right ) -1 \right ) -\ln \left ( 1+y \left ( x \right ) \right ) +\ln \left ( y \left ( x \right ) \right ) =0 \right \} \] 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),'implicit')
Maple raw output
_C1+ln(x^2+1)-ln(y(x)-1)-ln(1+y(x))+ln(y(x)) = 0