ODE
\[ \left (x^2+1\right ) y(x) y'(x)+x \left (1-y(x)^2\right )=0 \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.0160068 (sec), leaf count = 47
\[\left \{\left \{y(x)\to -\sqrt {e^{2 c_1} \left (x^2+1\right )+1}\right \},\left \{y(x)\to \sqrt {e^{2 c_1} \left (x^2+1\right )+1}\right \}\right \}\]
Maple ✓
cpu = 0.007 (sec), leaf count = 18
\[ \left \{ -{\it \_C1}\,{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-{\it \_C1}-1=0 \right \} \] Mathematica raw input
DSolve[x*(1 - y[x]^2) + (1 + x^2)*y[x]*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[1 + E^(2*C[1])*(1 + x^2)]}, {y[x] -> Sqrt[1 + E^(2*C[1])*(1 + x^2)]}}
Maple raw input
dsolve((x^2+1)*y(x)*diff(y(x),x)+x*(1-y(x)^2) = 0, y(x),'implicit')
Maple raw output
-_C1*x^2+y(x)^2-_C1-1 = 0