4.20.15 $xy(x)y^{\prime }(x{)}^2+(y(x)+x)y^{\prime }(x)+1=0$

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

[_quadrature]

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

Mathematica ✓
cpu = 0.00559688 (sec), leaf count = 53

$$\{\{y(x)\to -\sqrt{2}\sqrt{c_1-x}\},\{y(x)\to \sqrt{2}\sqrt{c_1-x}\},\{y(x)\to c_1-\log (x)\}\}$$

Maple ✓
cpu = 0.007 (sec), leaf count = 23

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

DSolve[1 + (x + y[x])*y'[x] + 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] 
-> C[1] - Log[x]}}

Maple raw input

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

Maple raw output

y(x) = -ln(x)+_C1, y(x)^2-_C1+2*x = 0