##### 4.18.1 $$x y'(x)^2+y'(x)=y(x)$$

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

[_rational, _dAlembert]

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

Mathematica
cpu = 1.66511 (sec), leaf count = 38

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

Maple
cpu = 0.796 (sec), leaf count = 57

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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