ODE
\[ x \left (-x^2+y(x)^2+1\right ) y'(x)+y(x) \left (x^2-y(x)^2+1\right )=0 \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Book solution method
Homogeneous equation, special
Mathematica ✓
cpu = 0.0656861 (sec), leaf count = 83
\[\left \{\left \{y(x)\to \frac {-\sqrt {-4 c_1 x^2+4 c_1^2+x^2}-2 c_1 x+x}{2 c_1}\right \},\left \{y(x)\to \frac {\sqrt {-4 c_1 x^2+4 c_1^2+x^2}-2 c_1 x+x}{2 c_1}\right \}\right \}\]
Maple ✓
cpu = 0.091 (sec), leaf count = 116
\[ \left \{ \left ( \left ( y \left ( x \right ) \right ) ^{-2}- \left ( {x}^{2}-1 \right ) ^{-1} \right ) ^{-1}-{x\sqrt {-1+x}\sqrt {1+x}{\frac {1}{\sqrt {{\it \_C1}-2\, \left ( 1+x \right ) ^{-1}+2\, \left ( -1+x \right ) ^{-1}}}}}+{\frac { \left ( -1+x \right ) \left ( 1+x \right ) }{2}}=0, \left ( \left ( y \left ( x \right ) \right ) ^{-2}- \left ( {x}^{2}-1 \right ) ^{-1} \right ) ^{-1}+{x\sqrt {-1+x}\sqrt {1+x}{\frac {1}{\sqrt {{\it \_C1}-2\, \left ( 1+x \right ) ^{-1}+2\, \left ( -1+x \right ) ^{-1}}}}}+{\frac { \left ( -1+x \right ) \left ( 1+x \right ) }{2}}=0 \right \} \] Mathematica raw input
DSolve[y[x]*(1 + x^2 - y[x]^2) + x*(1 - x^2 + y[x]^2)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x - 2*x*C[1] - Sqrt[x^2 - 4*x^2*C[1] + 4*C[1]^2])/(2*C[1])}, {y[x] -> (x - 2*x*C[1] + Sqrt[x^2 - 4*x^2*C[1] + 4*C[1]^2])/(2*C[1])}}
Maple raw input
dsolve(x*(1-x^2+y(x)^2)*diff(y(x),x)+(1+x^2-y(x)^2)*y(x) = 0, y(x),'implicit')
Maple raw output
1/(1/y(x)^2-1/(x^2-1))+(-1+x)^(1/2)*(1+x)^(1/2)*x/(_C1-2/(1+x)+2/(-1+x))^(1/2)+1/2*(-1+x)*(1+x) = 0, 1/(1/y(x)^2-1/(x^2-1))-(-1+x)^(1/2)*(1+x)^(1/2)*x/(_C1-2/(1+x)+2/(-1+x))^(1/2)+1/2*(-1+x)*(1+x) = 0