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 = 0.112043 (sec), leaf count = 61

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

Maple
cpu = 0.423 (sec), leaf count = 35

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-{x}^{2}=0,y \relax (x ) =\sqrt {-{{\it \_C1}}^{2}+1}+\sqrt {{x}^{2}-1}{\it \_C1} \right \} \] 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[2*ArcTan[Sqrt[(-1 + x)/(1 + x)]] + I*C[1]]}, {y[x] -> x*Cos[2*Ar
cTan[Sqrt[(-1 + x)/(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),'implicit')

Maple raw output

y(x)^2-x^2 = 0, y(x) = (-_C1^2+1)^(1/2)+(x^2-1)^(1/2)*_C1