4.19.19 $(1-x^2)y^{\prime }(x{)}^2+4x^2+2xy(x)y^{\prime }(x)=0$

ODE
$$(1-x^2)y^{\prime }(x{)}^2+4x^2+2xy(x)y^{\prime }(x)=0$$ ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for $y$

Mathematica ✓
cpu = 0.349725 (sec), leaf count = 24

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

Maple ✓
cpu = 0.292 (sec), leaf count = 31

$$\{{\left(y\left(x\right)\right)}^2+4x^2-4=0,y\left(x\right)=-\mathit{\_}\mathit{C}\mathit{1}+\mathit{\_}\mathit{C}\mathit{1}{x}^2-{\mathit{\_}\mathit{C}\mathit{1}}^{-1}\}$$ Mathematica raw input

DSolve[4*x^2 + 2*x*y[x]*y'[x] + (1 - x^2)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (4 - 4*x^2 + C[1]^2)/(2*C[1])}}

Maple raw input

dsolve((-x^2+1)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+4*x^2 = 0, y(x),'implicit')

Maple raw output

y(x)^2+4*x^2-4 = 0, y(x) = -_C1+_C1*x^2-1/_C1