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 = 0.54138 (sec), leaf count = 57
\[\left \{\left \{y(x)\to -\sqrt {-x^2+c_1 x+1-\frac {c_1{}^2}{2}}\right \},\left \{y(x)\to \sqrt {-x^2+c_1 x+1-\frac {c_1{}^2}{2}}\right \}\right \}\]
Maple ✓
cpu = 1.229 (sec), leaf count = 133
\[\left [y \left (x \right ) = -\frac {\sqrt {-2 x^{2}+4}}{2}, y \left (x \right ) = \frac {\sqrt {-2 x^{2}+4}}{2}, y \left (x \right ) = \sqrt {-2 \sqrt {1-\textit {\_C1}}\, x +2 \textit {\_C1} -x^{2}-1}, y \left (x \right ) = \sqrt {2 \sqrt {1-\textit {\_C1}}\, x +2 \textit {\_C1} -x^{2}-1}, y \left (x \right ) = -\sqrt {-2 \sqrt {1-\textit {\_C1}}\, x +2 \textit {\_C1} -x^{2}-1}, y \left (x \right ) = -\sqrt {2 \sqrt {1-\textit {\_C1}}\, x +2 \textit {\_C1} -x^{2}-1}\right ]\] 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))
Maple raw output
[y(x) = -1/2*(-2*x^2+4)^(1/2), y(x) = 1/2*(-2*x^2+4)^(1/2), y(x) = (-2*(1-_C1)^(1/2)*x+2*_C1-x^2-1)^(1/2), y(x) = (2*(1-_C1)^(1/2)*x+2*_C1-x^2-1)^(1/2), y(x) = -(-2*(1-_C1)^(1/2)*x+2*_C1-x^2-1)^(1/2), y(x) = -(2*(1-_C1)^(1/2)*x+2*_C1-x^2-1)^(1/2)]