ODE
\[ x^2+2 x y(x) y'(x)+2 y(x)^2 y'(x)^2+y(x)^2-1=0 \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.452344 (sec), leaf count = 57
\[\left \{\left \{y(x)\to -\sqrt {c_1 x-\frac {c_1^2}{2}-x^2+1}\right \},\left \{y(x)\to \sqrt {c_1 x-\frac {c_1^2}{2}-x^2+1}\right \}\right \}\]
Maple ✓
cpu = 0.256 (sec), leaf count = 106
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+{\frac {{x}^{2}}{2}}-1=0,[x \left ( {\it \_T} \right ) ={1 \left ( \sqrt {-{{\it \_C1}}^{2}+1}\sqrt {{{\it \_T}}^{2}+1}-{\it \_C1}\,{\it \_T} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}],[x \left ( {\it \_T} \right ) =-{1 \left ( \sqrt {-{{\it \_C1}}^{2}+1}\sqrt {{{\it \_T}}^{2}+1}+{\it \_C1}\,{\it \_T} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] Mathematica raw input
DSolve[-1 + x^2 + y[x]^2 + 2*x*y[x]*y'[x] + 2*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[1 - x^2 + x*C[1] - C[1]^2/2]}, {y[x] -> Sqrt[1 - x^2 + x*C[1] - C[1]^2/2]}}
Maple raw input
dsolve(2*y(x)^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)-1+x^2+y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x)^2+1/2*x^2-1 = 0, [x(_T) = -((-_C1^2+1)^(1/2)*(_T^2+1)^(1/2)+_C1*_T)/(_T^2+1)^(1/2), y(_T) = _C1/(_T^2+1)^(1/2)], [x(_T) = ((-_C1^2+1)^(1/2)*(_T^2+1)^(1/2)-_C1*_T)/(_T^2+1)^(1/2), y(_T) = _C1/(_T^2+1)^(1/2)]