##### 4.16.34 $$y'(x)^2+x y'(x)+1=0$$

ODE
$y'(x)^2+x y'(x)+1=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $$x$$

Mathematica
cpu = 0.169742 (sec), leaf count = 83

$\left \{\left \{y(x)\to -\frac {x^2}{4}+\frac {1}{4} \sqrt {x^2-4} x-\log \left (\sqrt {x^2-4}+x\right )+c_1\right \},\left \{y(x)\to -\frac {x^2}{4}-\frac {1}{4} \sqrt {x^2-4} x+\log \left (\sqrt {x^2-4}+x\right )+c_1\right \}\right \}$

Maple
cpu = 0.041 (sec), leaf count = 63

$\left [y \left (x \right ) = -\frac {x^{2}}{4}-\frac {x \sqrt {x^{2}-4}}{4}+\ln \left (x +\sqrt {x^{2}-4}\right )+\textit {\_C1}, y \left (x \right ) = -\frac {x^{2}}{4}+\frac {x \sqrt {x^{2}-4}}{4}-\ln \left (x +\sqrt {x^{2}-4}\right )+\textit {\_C1}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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