##### 4.20.33 $$a-x^2-2 x y(x) y'(x)+y(x)^2 y'(x)^2+2 y(x)^2=0$$

ODE
$a-x^2-2 x y(x) y'(x)+y(x)^2 y'(x)^2+2 y(x)^2=0$ ODE Classiﬁcation

[_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]]

Book solution method
Change of variable

Mathematica
cpu = 0.664712 (sec), leaf count = 63

$\left \{\left \{y(x)\to -\sqrt {-\frac {a}{2}-x^2+4 c_1 x-2 c_1{}^2}\right \},\left \{y(x)\to \sqrt {-\frac {a}{2}-x^2+4 c_1 x-2 c_1{}^2}\right \}\right \}$

Maple
cpu = 1.234 (sec), leaf count = 145

$\left [y \left (x \right ) = -\frac {\sqrt {4 x^{2}-2 a}}{2}, y \left (x \right ) = \frac {\sqrt {4 x^{2}-2 a}}{2}, y \left (x \right ) = \sqrt {-2 \sqrt {a +2 \textit {\_C1}}\, x -\textit {\_C1} -x^{2}-a}, y \left (x \right ) = \sqrt {2 \sqrt {a +2 \textit {\_C1}}\, x -\textit {\_C1} -x^{2}-a}, y \left (x \right ) = -\sqrt {-2 \sqrt {a +2 \textit {\_C1}}\, x -\textit {\_C1} -x^{2}-a}, y \left (x \right ) = -\sqrt {2 \sqrt {a +2 \textit {\_C1}}\, x -\textit {\_C1} -x^{2}-a}\right ]$ Mathematica raw input

DSolve[a - x^2 + 2*y[x]^2 - 2*x*y[x]*y'[x] + y[x]^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[-1/2*a - x^2 + 4*x*C[1] - 2*C[1]^2]}, {y[x] -> Sqrt[-1/2*a - x^2
 + 4*x*C[1] - 2*C[1]^2]}}

Maple raw input

dsolve(y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+a-x^2+2*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = -1/2*(4*x^2-2*a)^(1/2), y(x) = 1/2*(4*x^2-2*a)^(1/2), y(x) = (-2*(a+2*_C
1)^(1/2)*x-_C1-x^2-a)^(1/2), y(x) = (2*(a+2*_C1)^(1/2)*x-_C1-x^2-a)^(1/2), y(x)
= -(-2*(a+2*_C1)^(1/2)*x-_C1-x^2-a)^(1/2), y(x) = -(2*(a+2*_C1)^(1/2)*x-_C1-x^2-
a)^(1/2)]