##### 4.18.2 $$x y'(x)^2+2 y'(x)-y(x)=0$$

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

[_rational, _dAlembert]

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

Mathematica
cpu = 24.0111 (sec), leaf count = 40

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

Maple
cpu = 0.742 (sec), leaf count = 63

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = x*exp(2*RootOf(-x*exp(2*_Z)+2*x*exp(_Z)-2*exp(_Z)+_C1+2*_Z-x))+2*exp(Roo
tOf(-x*exp(2*_Z)+2*x*exp(_Z)-2*exp(_Z)+_C1+2*_Z-x))]