##### 4.18.34 $$2 x y'(x)^2+(2 x-y(x)) y'(x)-y(x)+1=0$$

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

[_rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 2.48627 (sec), leaf count = 43

$\text {Solve}\left [\left \{x=\frac {\frac {1}{K[1]+1}+\log (K[1]+1)+c_1}{K[1]^2},2 x K[1]+\frac {1}{K[1]+1}=y(x)\right \},\{y(x),K[1]\}\right ]$

Maple
cpu = 1.263 (sec), leaf count = 146

$[y \left (x \right ) = -\left (-2 \left ({\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_C1} +{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )}-1\right )^{2}-2 \,{\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_C1} +{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )}+2\right ) {\mathrm e}^{-\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_C1} +{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )} x +{\mathrm e}^{-\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_C1} +{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )}]$ Mathematica raw input

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

Mathematica raw output

Solve[{x == (C[1] + (1 + K[1])^(-1) + Log[1 + K[1]])/K[1]^2, 2*x*K[1] + (1 + K[1
])^(-1) == y[x]}, {y[x], K[1]}]

Maple raw input

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

Maple raw output

[y(x) = -(-2*(exp(RootOf(-x*exp(3*_Z)+2*x*exp(2*_Z)+exp(_Z)*_C1+exp(_Z)*_Z-x*exp
(_Z)+1))-1)^2-2*exp(RootOf(-x*exp(3*_Z)+2*x*exp(2*_Z)+exp(_Z)*_C1+exp(_Z)*_Z-x*e
xp(_Z)+1))+2)*exp(-RootOf(-x*exp(3*_Z)+2*x*exp(2*_Z)+exp(_Z)*_C1+exp(_Z)*_Z-x*ex
p(_Z)+1))*x+exp(-RootOf(-x*exp(3*_Z)+2*x*exp(2*_Z)+exp(_Z)*_C1+exp(_Z)*_Z-x*exp(
_Z)+1))]