4.19.35 $x(1-x^2)y^{\prime }(x{)}^2-2(1-x^2)y(x)y^{\prime }(x)+x(1-y(x{)}^2)=0$

ODE
$$x(1-x^2)y^{\prime }(x{)}^2-2(1-x^2)y(x)y^{\prime }(x)+x(1-y(x{)}^2)=0$$ 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

$$\{\{y(x)\to x\cos (2{\tan }^{-1}\left(\sqrt{\frac{x-1}{x+1}}\right)+i{c}_1)\},\{y(x)\to x\cos (2{\tan }^{-1}\left(\sqrt{\frac{x-1}{x+1}}\right)-i{c}_1)\}\}$$

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

$$\{{\left(y\left(x\right)\right)}^2-x^2=0,y\left(x\right)=\sqrt{-{\mathit{\_}\mathit{C}\mathit{1}}^2+1}+\sqrt{x^2-1}\mathit{\_}\mathit{C}\mathit{1}\}$$ 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