##### 4.19.35 $$x \left (1-x^2\right ) y'(x)^2-2 \left (1-x^2\right ) y(x) y'(x)+x \left (1-y(x)^2\right )=0$$

ODE
$x \left (1-x^2\right ) y'(x)^2-2 \left (1-x^2\right ) y(x) y'(x)+x \left (1-y(x)^2\right )=0$ ODE Classiﬁcation

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

Book solution method
Change of variable

Mathematica
cpu = 0.299807 (sec), leaf count = 97

$\left \{\left \{y(x)\to x \cos \left (\frac {\sqrt {x^2-1} \tan ^{-1}\left (\sqrt {x^2-1}\right )}{\sqrt {x-1} \sqrt {x+1}}-i c_1\right )\right \},\left \{y(x)\to x \cos \left (\frac {\sqrt {x^2-1} \tan ^{-1}\left (\sqrt {x^2-1}\right )}{\sqrt {x-1} \sqrt {x+1}}+i c_1\right )\right \}\right \}$

Maple
cpu = 0.831 (sec), leaf count = 33

$\left [y \left (x \right ) = x, y \left (x \right ) = -x, y \left (x \right ) = \sqrt {-\textit {\_C1}^{2}+1}+\sqrt {x^{2}-1}\, \textit {\_C1}\right ]$ Mathematica raw input

DSolve[x*(1 - y[x]^2) - 2*(1 - x^2)*y[x]*y'[x] + x*(1 - x^2)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> x*Cos[(Sqrt[-1 + x^2]*ArcTan[Sqrt[-1 + x^2]])/(Sqrt[-1 + x]*Sqrt[1 + x
]) - I*C[1]]}, {y[x] -> x*Cos[(Sqrt[-1 + x^2]*ArcTan[Sqrt[-1 + x^2]])/(Sqrt[-1 +
 x]*Sqrt[1 + x]) + I*C[1]]}}

Maple raw input

dsolve(x*(-x^2+1)*diff(y(x),x)^2-2*(-x^2+1)*y(x)*diff(y(x),x)+x*(1-y(x)^2) = 0, y(x))

Maple raw output

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