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 = 0.033692 (sec), leaf count = 83

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

Maple
cpu = 0.029 (sec), leaf count = 63

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

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

Mathematica raw output

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

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