4.20.5 $y(x)y^{\prime }(x{)}^2-(xy(x)+1)y^{\prime }(x)+x=0$

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

[_quadrature]

Book solution method
No Missing Variables ODE, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0050977 (sec), leaf count = 52

$$\{\{y(x)\to -\sqrt{2}\sqrt{c_1+x}\},\{y(x)\to \sqrt{2}\sqrt{c_1+x}\},\{y(x)\to c_1+\frac{x^2}{2}\}\}$$

Maple ✓
cpu = 0.007 (sec), leaf count = 24

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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