4.19.2 $x^2{y}^{\prime }(x{)}^2-2xy(x)y^{\prime }(x)+y(x)(y(x)+1)-x=0$

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

[[_1st_order, _with_linear_symmetries], _rational]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.0332616 (sec), leaf count = 55

$$\{\{y(x)\to \frac{c_1^{2}x}{4}-i{c}_1\sqrt{x}+x-1\},\{y(x)\to \frac{c_1^{2}x}{4}+i{c}_1\sqrt{x}+x-1\}\}$$

Maple ✓
cpu = 0.244 (sec), leaf count = 22

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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